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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 +R +t +R +t +R +t +@0.1◦ +1.52 +0.01 +6.67 +0.02 +2.14 +0.01 +8.41 +0.09 +@0.5◦ +14.74 +0.25 +44.87 +0.48 +21.47 +0.36 +44.14 +1.96 +@1.0◦ +28.92 +0.96 +64.15 +1.80 +40.99 +1.40 +64.48 +6.48 +@3.0◦ +55.75 +5.85 +84.76 +9.60 +68.08 +9.18 +86.89 +24.00 +@5.0◦ +68.27 +10.94 +90.34 +17.71 +77.39 +17.58 +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. 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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 \ No newline at end of file diff --git a/09FLT4oBgHgl3EQfpS-5/content/tmp_files/load_file.txt b/09FLT4oBgHgl3EQfpS-5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6c9062d7aebc65c0f0e7deb73fb289ae0c60c9b --- /dev/null +++ b/09FLT4oBgHgl3EQfpS-5/content/tmp_files/load_file.txt @@ -0,0 +1,1075 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf,len=1074 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In this paper, we propose AdaSfM: a coarse- to-fine adaptive SfM approach that is scalable to large-scale and challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Finally, our approach uses a threshold-adaptive strategy to align all local reconstructions to the coordinate frame of global SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Extensive experiments on large-scale benchmark datasets show that our approach achieves state-of-the-art accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' INTRODUCTION Structure from Motion (SfM) is an important topic that has been studied intensively over the past two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Despite the impressive results from many existing works, SfM remains challenging in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The first challenge is outlier feature matches caused by the diversity of scene features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' texture-less, self-similar, non-Lambertian, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' These diverse features impose challenges in sparse feature extraction and matching which result in outliers that are detri- mental to the subsequent reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Incremental SfM [7], [8] is notoriously known to suffer from drift due to error accumulation, though is robust in handling outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The second challenge is sparse view graphs from some large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Incremental SfM is known to be ineffi- cient on large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Several works [16], [17], [18], [19] have been proposed to handle millions of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' These are divide-and-conquer SfM methods that deal with very large-scale datasets by grouping images into partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Each partition is processed by a cluster of servers that concurrently 1School of Computing, National University of Singapore, {chenyu, gimhee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='lee}@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='sg 2Segway-Ninebot Robotics Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=', Ltd, yuzihao@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='cn, songshu0905@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='com 3Navimow B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=', Ltd, tianning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='yu@rlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='segway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='com, jianming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='li@ninebot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='com Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' When combining with global SfM, our AdaSfM is more robust than traditional incremental SfM (tested on the public 4Seasons dataset [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' circumvents the memory limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' However, these methods [16], [17], [18], [19] are often limited to internet datasets or aerial images where the view graphs are very densely connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The dense connections in the view graph ensure that there are sufficient constraints between the graph par- titions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' An example of such a dataset is autonomous self-driving cars where the interval between consecutive images can be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' This augmented view graph is then used to estimate arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='12135v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='CV] 28 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The pipeline of our proposed SfM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Our method takes images and measurements from low-cost sensors as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The view graph is built after feature matching and refined by the result of global SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The absolute poses from the global SfM are used as priors in the subsequent local SfM process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The final reconstruction result is merged into the global SfM reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' the global poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Consequently, we obtain a coarse scene structure and camera poses, where the latter can be used to filter wrong feature matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We define separators as images that connect different sub-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We illustrate the pipeline of our method in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We evaluate our method extensively on large-scale chal- lenging scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Experimental results show that our AdaSfM is adaptive to different scene structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Furthermore, we achieve better robustness and comparable efficiency in com- parison to existing state-of-the-art SfM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' RELATED WORK Incremental SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [7] apply preconditioned conjugate gradient [25] to accelerate large-scale BA [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The drift problem is alleviated in [27] with a re-triangulation (RT) step before global BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In addition to the RT before BA [27], RT is also performed after BA in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Global SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Govindu [29] represents rotations by lie- algebra, and global rotations and global positions are es- timated simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' To solve the TA problem, Wilson et al [33] project relative translations onto the 1D space to identify outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Relative translations that are inconsistent with the translation directions that have the highest consensus are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' A nonlinear least-squares problem is then solved to get the global positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [34] relax the scale constraints of [33] to linear scale factors, and the convex linear programming problem is solved by ADMM [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' ¨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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' By minimizing the sin θ between two relative translations, Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [36] improve the insensitivity to narrow baselines of TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The robustness of TA is also improved in [36] by incorporating global rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Hybrid SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [37] obtain orientations by RA and then register camera centers incrementally with the perspective-2-point (P2P) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Bhomick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [16] propose to divide the scene graph, where the graph is built from the similarity scores between images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Feature matching and local SfM can then be executed in parallel and local reconstructions are merged [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [18], [19] adopt a similar strategy to divide the scene and the graph is constructed after feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The relative poses are collected after merging all local incremental reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Based on [18], Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [17] find the minimum spanning tree (MST) to solve the final merging step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The MST is constructed at the cluster level, and the most accurate similarity transformations between clusters are given by the MST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Locher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [38] filtered wrong epipolar geometries by RA before applying the divide- and-conquer method [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' [39] use a visual- inertial navigation system (VINS) [40] to first estimate the camera trajectories with loop detection and loop closure [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Images are then divided into sequences according to timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' However, [39] requires two carefully designed systems: one for VINS with loop detection and the other for SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Loop detection is also a challenge in real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The absolute camera poses project 3D points X = {Xk} from the world frame to the camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The camera centers are denoted by {Ci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We define the view graph as G = {V, E}, where V denotes the collection of images and E denotes the two view geometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' the relative poses and inlier matches between the image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Additionally, the keypoints and the normalized keypoints after applying the intrinsic matrix K are denoted by u and ˆu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' COARSE GLOBAL TO FINE INCREMENTAL SFM In this section, we introduce our method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' IV-A, we introduce our global SfM that can effectively cope with outliers in challenging scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' A refinement step is also introduced to remove outlier matches after global SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (1) can be solved robustly and efficiently by [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We then obtain the absolute camera positions by solving the translation averaging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2) The view graph is solvable only when the parallel rigid graph condition [12] is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' ∥tij∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Furthermore, the state-of-the-art method to establish the solvability of a view graph is only limited to 90 nodes [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We do not augment the relative rotations because they are more accurate from the image-based two-view geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' To circumvent this problem, we only use the relative poses where the time difference is below a threshold ϵT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We can then further solve the translation averag- ing problem below: arg min ˆ Ci,i∈V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' sij,(i,j)∈Eaug � (i,j)∈Eaug ∥sij( ˆCi − ˆCj) − R⊤ j tij∥, (3) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' sij ≥ 0, ∀(i, j) ∈ Eaug;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' � i∈V ˆCi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (3) can be solved efficiently and robustly under the l1- norm by collecting all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Note all the relative translations are normalized in Eaug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 3 shows our global SfM result by solving (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' After translation averaging, we triangulate the 3D points and perform an iterative global bundle adjustment to refine camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Besides, according to [28], tracks are redundant for optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Therefore, we can reduce the computation and memory burden with fewer tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Note that for numerical stability during optimization, the coordinates are normalized after each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Comparison of global SfM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Results from [12] (left) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (3) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Red and black colors respectively denote vehicle trajectories and sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (ˆtij, ˆRij) are computed from ( ˆRi, ˆRj) and (ˆti,ˆtj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The true matches ˆu′ ↔ ˆu must satisfy the check on the total point-to-epipolar line distance [22] over the two views, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' d⊥(ˆu, Eˆu′) + d⊥(ˆu′, Eˆu) ≤ ϵM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (4) d⊥(x, l) gives the shortest distance between a point x and a line l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The epipolar lines on the two images are given by l = Eˆu′ and l′ = Eˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' ϵM is the threshold for the check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The effectiveness of global SfM to filter wrong matches can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We build a pseudo ground truth by COLMAP [8] to evaluate the accuracy of the global SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The ratio test is performed after NN by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 4 shows the inlier ratio distribution after NN+RANSAC and matches refinement with relative poses obtained from global SfM and incremental SfM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' I gives the relative pose estimation AUC of NN+RANSAC and global SfM with respect to incremental SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' It can be seen that our coarse global SfM can obtain comparable accuracy to COLMAP [8] in the refinement of the matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Inlier ratio distribution of NN+RANSAC, global SfM and incremental SfM (ground truth) on the 711 (left) and B6 (right) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' AUC NN+RANSAC Global SfM NN+RANSAC Global SfM R t R t R t R t @0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='1◦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='67 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='95 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='00 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Finer Parallel Incremental SfM Although we have obtained the absolute camera poses by global SfM, these coarse poses are not accurate enough for localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' To improve the accuracy, we propose to refine the camera poses and scene structure with the divide-and- conquer incremental SfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 1) Adaptive Graph Partition: Existing approaches [18], [17] used a cut-and-expand schema to create overlapping areas between partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' However, these approaches have two main drawbacks: : 1) The overlapping areas are not enough for final merging when the view graph becomes too sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' This can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5(a), only images {3, 7, 8, 9, 16, 17, 18, 19, 20} can be used to create the over- lapping areas (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' However, these separator images are insufficient to compute the similarity transformations for merging all local reconstructions due to the sparsity of the view graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2) Graph cut tends to separate partitions along edges with weak associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' This means the separators are often weakly constrained during reconstruction and thus their poses might not be accurate enough during reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We propose a flood-fill graph partition algorithm to over- come the above-mentioned disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We refer to the added nodes in each cluster after an expansion operation as a layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The separators are collected to form a layer after the graph cut on the complete view graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5(a) shows examples of the separators marked green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We have separators S1 = {{3, 7, 8}, {9, 16, 17}, {18, 19, 20}} in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We then collect all the adjacent images of every separator for each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Adjacent images are sorted in descending order according to the weights of the edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' the number of inlier matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5(b) shows that the separators S2 = {{9, 20}, {8, 18}, {8, 16}} at the second layer after traversing all separators in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The expansion step is repeated until the number of overlapping images reaches the overlapping threshold τot (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 30%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5(c) shows the separators S3 at the third layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2) Local Incremental SfM: We perform incremental SfM in parallel after graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' a) Image Registration: We follow [8] for the two-view initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' For each candidate image i, we first use the P3P [44] to compute the initial pose Pp3p i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' However, images can be registered wrongly due to wrong matches or scene degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We propose to also compute the image pose Pgb i = [Rgb i | tgb i ] using ˆPglobal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Finally, we select the pose which has the most inliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='00 inlier ratioGround Truth 60k GlobalSfM NN ransac 50k 40k 30k 20k 10k ok 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='00 inlier ratio(a) Initial graph cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (b) The 1st graph expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (c) The 2nd graph expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Pipeline of adaptive flood-fill graph partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In the view graph, nodes are denoted by blue circles, edges are denoted by blue solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Separators are marked by green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Vehicle trajectories of different threshold trials when merging sub-reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The last figure is obtained by our method which starts from an initial inlier threshold τinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Others are the results of using a fixed threshold during the alignment to merge all local reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' where Π(·) reprojects a 3D point back to the image plane, d∠(·) denotes the angle between two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Note that we do not make the hard constraint to force the translation part of ˆP−1 ij Pij to be a zero-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Instead, we use d∠(tij,ˆtij) = d∠(Ci−Cj, ˆCi− ˆCj) to constrain the translation direction of camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' This is because the absolute positions obtained from global SfM are not sufficiently accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The difficulties in estimating a reliable similarity transformation are due to 1) The existence of outliers in registered camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Although the outliers can be identified by RANSAC [46], the threshold that indicates outliers is hard to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' This is due to the loss of the absolute scale of the real world in SfM without additional information such as GPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' It indicates that the optimal outlier threshold varies for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2) The estimated similarity transformation can overfit wrongly with insufficient sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' To tackle the first issue, we propose an adaptive strategy to determine the inlier threshold τinlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Given an initial inlier threshold τinit, we first estimate the similarity transformation by RANSAC [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We then compute the inlier ratio rinlier and increase the inlier threshold if rinlier < rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Furthermore, we decrease the threshold if rinlier ≥ rmax to prevent the threshold from becoming too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' A large threshold allows more outliers to be falsely selected and thus harming the similarity transformation estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The second issue can be solved easily within our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We set the coordinate frame of the global SfM as the reference frame, and align each local SfM into the reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We also show the effectiveness of the algorithm to merge local reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' When zooming in, we can observe that our adaptive strategy perfectly closed the loop while other fixed threshold trials failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' EXPERIMENTAL RESULTS In this section, we perform extensive experiments to demonstrate the accuracy, efficiency, and robustness of our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Implementation Details We use HFNet [48] as the default feature extractor and use the NN search for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We assume cameras are pre-calibrated and use the ceres-solver [49] for bundle adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We did not compare our method against [39], as VINs [40] fails to find the right loops in our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' All methods are run on the same computer with 40 CPU cores and 96 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Evaluation Datasets: We evaluate our method on our self- collected outdoor datasets and the 4seasons [15] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Our self-collected datasets are collected by low-speed au- tonomous mowers, of which the running environments have many plants and texture-less areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The 4seasons dataset is a cross-season dataset that includes multi-sensor data such as IMU, GNSS, and stereo images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' See our attached video for a more qualitative and quantitative evaluation of the 4Seasons dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='5 Tinlier = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='0 Tinlier = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='5 Tinlier = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='0 Tinit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='0Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Vehicle trajectories after match refinement on B6 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (b), the visual results are respectively reconstructed without (left) and with (right) match refinement in each sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (c) shows some of the wrong matching pairs that are filtered by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Dataset N COLMAP [8] GraphSfM [17] Ours(Global SfM) Ours(Global+Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=') 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='47 597,171 48,491 540,711 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='38 88,896 (×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='7 ↑) 48,758 521,080 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='51 5,177 48,694 540,942 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='66 105,163 (×5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='7 ↑) 711 29,619 27,175 303,352 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='64 160,322 29,618 259,292 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='46 33,514 (×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='8 ↑) 29,629 249,673 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='86 3,499 29,619 256,495 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='61 38,682 (×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='1 ↑) yht 7,472 7,470 90,437 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='16 20,428 6,709 78,659 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='17 7,526 (×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='7 ↑) 7,472 132,167 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='67 524 7,472 108,711 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='43 9,778 (×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='1 ↑) A4 5,184 5,132 33,694 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='69 18,104 4,285 28,726 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='55 12,670 (×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='4 ↑) 5,184 24,193 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='59 1,349 5,184 34,007 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='43 6,924 (×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='6 ↑) Htbd 14,651 14,645 231,870 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='30 56,888 14,645 232,441 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='37 17,187 (×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='3 ↑) 14,646 190,904 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='47 1,523 14,646 238,035 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='36 16,852 (×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='4 ↑) jy1 32,484 32,463 534,117 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='44 346,161 32,466 536,331 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='52 28,673 (×12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='1 ↑) 32,484 463052 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='12 3,077 32,466 621,437 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='53 33,555 (×10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='3 ↑) TABLE II COMPARISON OF RUNTIME AND ACCURACY ON REAL-WORLD DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' FOR RUNTIME T (SECONDS), THE FIRST, SECOND AND THIRD THE BEST RESULTS ARE HIGHLIGHTED IN COLOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The point-to-epipolar line distance is ϵM = 4 px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Besides, we set the overlapping ratio τot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='3 in the graph partition, vp = 10 for an image to be a candidate to register, and rmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='7, rmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='9, τinit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='0, αinc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='2, αdec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='1 in global alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' How Matching Refinement Saves SfM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In addition to running our experiments on HFNet, we also do evaluations on different trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We first show the reconstruction results conducted on a challenging scene in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 7, which is difficult for visual methods to identify the wrong feature matches due to specular issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We use two different combinations of methods for feature extraction and matching in each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In the first combi- nation, we use HFNet [48] for feature extraction and NN search for feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In the second combination, we use Superpoint [52] for feature extraction and Superglue [53] for feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Both settings use RANSAC [46] to remove matching outliers that do not satisfy the point-to- epipolar line constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In each sub-figure, the left and right images are the results without and with matching refinement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' For Superpoint + Superglue, the state-of-the-art methods respectively on feature extraction and matching, also fails on the dataset without refining matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Qualitative Evaluation on Real-World Datasets We evaluated our full pipeline on several outdoor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' As shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' II, our method shows the most number of registered images in almost all the datasets, while [17] shows the least number of registered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Interestingly, GraphSfM [17] is almost 1× slower than our method on the A4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Our explanation is validated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' II where GraphSfM [17] recovers only 4,235 poses out of 5,184 images, which is almost 20% less than our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a robust SfM method that is adaptive to scenes in different scales and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': 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networks,” in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 4937–4946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Adaptive Flood-Fill Graph Partition Algorithm The pseudo-code of our adaptive flood-fill graph partition algorithm is given in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2: while τor < τot do 3: Update separators Vs = {Vs 0, · · · , Vs K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 4: Edges Edis := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5: for k ∈ [0, K] do 6: Edis k = E − Ek and Edis k contains Vs k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 7: Edis+ = Edis k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 8: Sort Edis by descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 9: for Edge e ∈ Edis do 10: Select a partition Gk contains one of the nodes in e and has the smallest size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 11: Add e to Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 12: Update τor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Adaptive Global Alignment Algorithm The pseudo-code of our adaptive global alignment algo- rithm is given in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 5: Compute sim3 by τinlier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 6: Compute rinlier by sim3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 7: if rinlier < rmin then 8: τinlier := τinlier + αinc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 9: else if rinlier ≥ rmax then 10: τinlier := τinlier − αdec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Visualization Results on Self-Collected Dataset The qualitative visualization results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Moreover, GraphSfM [17] fails to correctly merge the sub-reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The misalignment can be observed from the zoom-in areas of side-view images, which further validates the robustness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Ablations of Augmented View Graph We present more ablation of the augmented view graph on the 4Seasons dataset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' More visualization results on this dataset can be seen in our attached video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Quantitative Results on 4Seasons dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We present the quantitative results on the 4Seasons dataset in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The 4Seasons dataset provides ground truth camera poses and trajectories from VI-Stereo-DSO [50], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' The sensor data contain IMU, GNSS, and stereo images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' In our experiment, we do not use the GNSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' We strongly recommend readers refer to [15] for more details about the challenged dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' As is expected, Our method outperforms COLMAP by a large margin in terms of both accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' As the two sequences contain severe motion blur and tunnels in images, which makes them very challenging to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' However, our method is also robust to these scenes since it can robustly fuse different sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (a) Qualititve comparison on the 711 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (b) Qualititve comparison on the A4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' (c) Qualititve comparison on the high free dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Reconstruction comparisons on our self-collected dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' From left to right are the input images, top-view reconstruction, and side-view reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' COLMAP SJnoCOLMAI s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='inoCOLMAP sJIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Ablations of our augmented view graph on the 4Seasons dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': 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+page_content='32 9,426 122,746 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='68 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='96 9,426 204,289 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='91 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content='29 TABLE III COMPARISON OF RUNTIME AND ACCURACY ON THE 4SEASONS DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} +page_content=' Global SfM from raw view graph gran' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FLT4oBgHgl3EQfpS-5/content/2301.12135v1.pdf'} diff --git a/39FAT4oBgHgl3EQfEhxj/content/2301.08422v1.pdf b/39FAT4oBgHgl3EQfEhxj/content/2301.08422v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..863b61b6d3314c5c463bbdd6e114c75ef29bba9f --- /dev/null +++ b/39FAT4oBgHgl3EQfEhxj/content/2301.08422v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b9ea56b7095d238958fdc2250cc92d4c70547ffbd4e90730d69f7ba3cdf28f05 +size 31803993 diff --git a/3NE1T4oBgHgl3EQf5wVj/content/tmp_files/2301.03515v1.pdf.txt b/3NE1T4oBgHgl3EQf5wVj/content/tmp_files/2301.03515v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cffb41e47bca051ca4f109a8785591e09e80f31d --- /dev/null +++ b/3NE1T4oBgHgl3EQf5wVj/content/tmp_files/2301.03515v1.pdf.txt @@ -0,0 +1,1635 @@ +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 − Δ𝑖𝑒−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. 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L., 1996, Solar System Research, 30, 480 +Wiegert P., Brown P., 2004, Earth Moon and Planets, 95, 19 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–10 (2022) + diff --git a/3NE1T4oBgHgl3EQf5wVj/content/tmp_files/load_file.txt b/3NE1T4oBgHgl3EQf5wVj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3121fbcdc410ab0845547bc6d5ed715dd60005a0 --- /dev/null +++ b/3NE1T4oBgHgl3EQf5wVj/content/tmp_files/load_file.txt @@ -0,0 +1,1611 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf,len=1610 +page_content='MNRAS 000, 1–10 (2022) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 Identifying meteorite droppers among the population of bright ’sporadic’ bolides imaged by the Spanish Fireball Network during the spring of 2022 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Peña-Asensio,1,2★ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Trigo-Rodríguez,2,3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Rimola,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Corretgé-Gilart,4 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We investigate their origins in the Solar System and their dynamic association with parent bodies and meteoroid streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Employing our Python pipeline 3D-FireTOC, we reconstruct the atmospheric trajectory utilizing ground- based multi-station observations and compute the heliocentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In addition, we applied an ablation model to estimate the initial and terminal mass of each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Using a dissimilarity criterion and propagating backward in time, we check the connection of these meteoroids with known complexes and near-Earth objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We also calculate if the orbits are compatible with recent meteoroid ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We identify two meteorite-producing events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' however, the on-site search was unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Murad & Williams 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Trigo-Rodríguez 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' These small bodies are fragments pro- duced by the catastrophic disruption or collisions of comets, aster- oids, or even impacts on planets (Chapman 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Tóth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Gritsevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Trigo-Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Vaubaillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Some of these meteoroid streams have Earth-intersecting orbits, so they are generally repeated in an- nual cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='pena@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='cat, eloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='peas@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='com complex (Olsson-Steel 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Bottke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Pauls & Gladman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Brož 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Koschny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Silber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' This phenomenon is known as a meteor and is called a fireball or bolide if its magnitude is greater than that of Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Halliday 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Madiedo & Trigo-Rodríguez 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Borovička et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Trigo-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Unfortunately, the weather during winter and spring © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='03515v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='EP] 9 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Location of the fireball observation points involved in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Station Name Long (◦) Lat (◦) Alt (m) A Alpicat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5568 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6676 252 B Barx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3041 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0146 336 C Benicàssim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0386 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0342 15 D Calar Alto 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='549 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2212 2152 E Cebreros 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3693 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4541 700 F Corbera 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8906 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4092 501 G Estepa 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8766 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2914 537 H GranTeCan 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8919 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='7567 2267 I La Murta 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6756 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0967 469 J Monfragüe 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0108 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='7736 411 K Morata de Jalón 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4821 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='474 415 L Olocau 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5363 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6744 225 M Playa Blanca 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8241 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8747 10 N Puertollano 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1129 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='7032 697 O Sant Mateu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1758 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='465 349 is usually not helpful for fireball monitoring and clouds generally prevent detailed trajectory reconstruction and strewn-field estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In section 2, we first outline the SPMN network’s current in- frastructure that has allowed recording these events with multiple stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We also mention the methodology applied for fireball anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In section 3, we describe the results of the atmospheric flight reconstruction, terminal mass prediction, and heliocentric orbit cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In section 4, we analyze the dynamic associations with parent bodies, near-Earth asteroids and comets, and minor and major meteoroid streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In addition, we examined the compatibility of these events being recently ejected meteoroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Finally, we discuss the results in section 5 and offer our conclusions in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In addition, three forward-scatter detectors monitor radio meteors (Trigo-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2021b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The integrator is based on the RADAU-15 developed in Everhart (1985) and has a high performance resolving close encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' For most cases, we performed the astrometric calibration by solv- ing the polynomial modification of Borovička (1992) proposed by Bannister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (2013), which exhibits a better convergence while ensuring a very excellent level of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' To achieve the best fit, we use a simplicial homology global optimization algorithm to find the absolute minimum (Endres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' For recordings with suf- ficient background stars, we apply the method proposed in Borovicka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (1995), which produces even lower errors down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='01◦ for azimuth and elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' All calibrations are also cross-checked with the quadratic model described in Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In this sense, we use 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='18 as a cut-off for 𝐷𝐷 (Galligan 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In addition, we evaluate if the separation of the meteoroid from its possible parent body could have occurred in relatively short timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' In this way, we can observe if the velocity change is compatible with typical collisional ejection processes between small bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' It is used to classify planet-crossing objects, usually, as Jupiter-family comets (JFCs) if 2 < 𝑇𝑗 < 3 and asteroidal when 𝑇𝑗 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' standard atmosphere 1976 (Bronshten 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' This parameter is typically used to mechanically characterize the meteoroid and to classify the material regarding the bulk density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Gritsevich 2008, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Gritsevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Turchak & Gritsevich 2014): 𝐹𝑖(ℎ𝑖, 𝑣𝑖, 𝛼, 𝛽) = 2𝛼𝑒−ℎ𝑖 − Δ𝑖𝑒−𝛽, (2) with Δ𝑖 = 𝐸𝑖(𝛽) − 𝐸𝑖(𝛽𝑣2 𝑖 ), 𝑖 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=', 𝑛, where 𝐸𝑖(𝑥) = ∫ 𝑥 −∞ 𝑒𝑡𝑑𝑡 𝑡 𝑑𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' These adimensional parameters are defined as 𝛼 = 1 2𝑐𝑑 𝜌0ℎ0𝑆0 𝑀0 sin 𝛾 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (3) and 𝛽 = (1 − 𝜇) 𝑐ℎ𝑣2 0 2𝑐𝑑𝐻∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (4) where 𝑐𝑑 is the drag coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 𝜌0 is the atmospheric density at sea level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' ℎ0 is the scale height for a homogeneous atmosphere and 𝛾 is the slope of the fireball to the local horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 𝑀0 is the meteoroid mass before impacting the top of the atmosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 𝜇 is the dimensionless shape change parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 𝑐ℎ is the heat transfer coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 𝑣0 is the entry velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' and 𝐻∗ is the sublimation heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' These parameters allow properly describing the atmospheric flight and estimating the meteor fate based on the so-called 𝛼 − 𝛽 criterion (Sansom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' From the aerodynamic strength values, we assign a mete- oroid bulk density based on Chyba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (1993): cometary if 𝑠 < 105 𝑃𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' carbonaceous if 105 𝑃𝑎 < 𝑠 < 106 𝑃𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' rocky if 106 𝑃𝑎 < 𝑠 < 107 𝑃𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' and rocky-iron if its aerodynamic strength is greater than 107 𝑃𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Each station recorded the events in a single shot, except for the grazing meteoroid SPMN080322, which moved out of the field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Therefore, we had to combine the recordings from two cameras to obtain the complete luminous trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Figure 1 shows a composite of overlapping images of some of the events recorded and analyzed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The beginning and ending position, distance flight, and direction of the luminous phase for each event are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The initial heights range from ∼ 120 to 83 km and terminal heights (before starting the dark flight) range from ∼ 80 to 13 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' As expected, the azimuth and slope have a random distribution, with the average slope being around 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Note that the slope is measured with respect to the local horizon, 0◦ corresponding to a fully grazing meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Selection of blended frames of some of the events analyzed in this work: a) SPMN090322C from Calar Alto by José M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Serna García, b) SPMN060222 from Corbera, c) SPMN080222B from Barx, d) SPMN220222 from Alpicat, e) SPMN180222 from Estepa, and f) SPMN110222 from Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Recorded fireballs with the beginning and ending position, flight distance traveled, and direction of the atmospheric flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='324±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='011 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='848±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='004 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Sinusoidal projection of the geocentric (diamond) and apparent (gray cross) radiants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Radiant pairs are connected with a light blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Geocentric radiants are color-coded according to their geocentric velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Using the height at which the brightest flare occurs, the air density, and the velocity at that point, we calculate the aerodynamic strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' To obtain the ballistic coefficient and the mass loss parameter, we assume an aerodynamic drag coefficient of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3 and a shape change coefficient of 2/3 (Gritsevich & Koschny 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' All the computed parameters are shown in Table 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Two meteoroids penetrate up to ∼ 30 and 13 km altitude starting the dark flight at a velocity of ∼ 8 and 20 km/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Unfortunately, a field search campaign was prepared but no fragments were recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The computed osculating orbital elements at the time of impact of the analyzed fireballs are compiled in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' As expected, the orbits tend to be of low inclination, with the exception of SPMN090322B which has an inclination of 122◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' None of the meteoroids had close encounters with the Moon prior to the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Distribution of the 15 fireballs analyzed over the Spanish territory during February and March 2022 according to the 𝛼 − 𝛽 criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We assume 𝜇 = 2/3 for all meteoroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' From the time evolution of the parent body dissimilarity criterion, we found some dynamic associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 5 out of 15 events, that is, about 30% of the bright fireballs, are below the cut-off for at least 5,000 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 4 events would be associated with minor showers (∼27%) and 1 fireball associated with a near- Earth asteroid (∼7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The low density and high porosity of the latter are directly related to their aerodynamic strengths (Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Cometary streams typically produce centimeter-sized projectiles causing fireballs with disruptive flares, and multiple sudden brightness increases or a catas- trophic final flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Recorded fireballs with aerodynamic strength, ballistic coefficient, mass loss parameter, pre-atmospheric diameter, pre-atmospheric mass, and terminal mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' SPMN code s (kPa) 𝛼 𝛽 D (cm) M0 (g) M𝑡 (g) 060222 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='7)·102 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='16 <1 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='08 192±7 106±20 100322 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='14)·103 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5)·103 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8)·102 120322 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='26 82±22 140±34 12±4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1)·103 <1 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Recorded fireballs with right ascension and declination of the radiant, apparent, geocentric, and heliocentric velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 106±4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8±1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='26 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='10 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='26 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='23 7±34 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='15 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='06 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='18 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='09 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Recorded fireballs with semi-major axis, eccentricity, inclination, perihelion distance, argument of the perihelion, ascending node, and Tisserand parameter (referred to the J2000 equinox).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Uncertainty for the ascending node is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0001◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' SPMN code a (au) e i (◦) q (au) 𝜔 (◦) Ω (◦) T 𝑗 060222 11±5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='04 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='892±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='011 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='823±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='011 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8516 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='21 080222A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='938±0.' metadata={'source': 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aerody- namic strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' This, in turn, allows for deducing the bulk properties of their meteoroid stream (Kresak 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Trigo-Rodríguez & Llorca 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Even in such circumstances, it is remarkable that the sporadic con- MNRAS 000, 1–10 (2022) Meteorite dropper spring 2022 7 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' SPMN code Association D𝑚𝑖𝑛 t𝐷 (y) S𝑚𝑖𝑛 (au) V𝑆,𝑚𝑖𝑛 (km/s) V𝑚𝑖𝑛 (km/s) 060222 𝜌 Geminids 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='9 080222B Southern 𝛿 Leonids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='018 8720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 110222 𝜔 Cassiopeiids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='101 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 090322C March Cassiopeiids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='084 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='34 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8 100322 𝜓 Draconids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='106 2080 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='590 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='595 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='600 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='605 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='610 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='615 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='620 Semi-major axis (au) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 Inclination (°) 1,000 clones heatmap t=0 year (impact) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='98 Semi-major axis (au) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 Inclination (°) 1,000 clones heatmap t=-10,000 year Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Typical heatmap of the inclination and semi-major axis distribution of the 1,000 clones for the SPMN010322A in the Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The top figure corresponds to the time of impact (t=0 year) without Earth- Moon gravitational focusing correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The bottom figure corresponds to the end of the backward orbital integration (t=-10,000 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' tribution is not dominant at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We found a very significant percent- age of bright fireballs dynamically associated with minor showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Some re- gions even undergo more pronounced decoherence than others due to the gravitational influence of the Earth-Moon system or nearby planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The minimum ejection velocities calculated to produce the me- teoroid orbit from the parent body have a standard deviation range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='16 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 km/s (with an average standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 km/s) for the studied events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' we just note it as a feasible possibility due to the usual disruption behavior of crumbling asteroids and comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' For example, percentages up to 80% between November and January were already reported belonging to meteor showers (Rao & Murthy 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' On the other hand, among the 2,401 records studied by Lindblad (1971), apparently, 37% were associated with meteoroid streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' A similar percentage (41%) was found by Southworth & Hawkins (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Of the orbits analyzed by Jacchia & Whipple (1961), 65% were linked to a meteor shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Regarding the Meteorite Observation and Recovery Project (MORP) database, 37% of the fireballs could be associated with meteoroid stream (Halliday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Recent studies also show large percentages of meteors associated with meteor showers, for example, MNRAS 000, 1–10 (2022) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Evolution of the dissimilarity function 𝐷𝐷 of the 15 meteoroids with their most favorable candidates during the orbital backward integration over 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The 1,000 clones of each event are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1 SPMN060222 - p Geminids SPMN080222A- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 SPMN110222 - w Cassiopeiids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1 SPMN140222B - March Cassiopeids SPMN180222 - Southern 6 Leonid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 SPMN220222 - 209 Northern α Leonids SPMN010322A - 2019 CV2 SPMN010322B - 2017 FM91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 - B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 7 SPMN090322B - 72 Ophiuchids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1 SPMN080322A - 2007 DZ40 SPMN080322B - February Hydrids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='6 2222727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='4 D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='1 SPMN090322C - March Cassiopeids SPMN100322 - Draconids SPMN120322 - 入 Leonids 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (2020) and 35% in Drolshagen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Re- garding superbolides detected from space, 23% could be associated with meteoroid streams or near-Earth objects (Peña-Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Ground-based multi-station recordings were possible thanks to the ever-increasing atmospheric volume monitored by the SPMN network throughout Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' We reported 15 bright bolides in February and March, two of them being potential meteorite dropper events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Nearby meteoroid streams can be efficient producers of large projectiles as they account for the ∼27% of the fireballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Near-Earth objects may be a greater source of impact risk than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' These findings support the idea that certain meteoroid streams associated with comets or asteroids may represent a short-term im- pact hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' The relevance of associations also reverts in outreach, as we can quickly inform the public about the origin of the fireballs reported by eyewitnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 865657) for the project “Quantum Chemistry on Interstellar Grains” (QUANTUMGRAIN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' JMT-R and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='P-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' acknowledge finan- cial support from project PID2021-128062NB-I00 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' AR acknowledge financial support from the FEDER/Ministerio de Ciencia e Innovación – Agen- cia Estatal de Investigación (PID2021-126427NB-I00, PI: AR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' AR is indebted to DIUE (project 2017SGR1323).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Cebreros #AMS81 ESA Ground station belongs to the AllSky7 fireball monitoring project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Serna, Rainer Kresken, Pablo Ramirez Moreta, Carlos Alcaraz, Antonio J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' Robles, Ramón López, Agustín Núñez, José A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQf5wVj/content/2301.03515v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': 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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. +[1] F. Hassler, A. R. 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B 102, 054507 (2020). +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 + diff --git a/3dE2T4oBgHgl3EQfjgcV/content/tmp_files/load_file.txt b/3dE2T4oBgHgl3EQfjgcV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27d080a4704341ef38c07cde6d7fedaccce3701b --- /dev/null +++ b/3dE2T4oBgHgl3EQfjgcV/content/tmp_files/load_file.txt @@ -0,0 +1,845 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf,len=844 +page_content='Ballistic surface channels in fully in situ defined Bi4Te3 Josephson junctions with aluminum contacts D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rosenbach,1, 2, ∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Jalil,3, 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Schmitt,1, 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Bennemann,3, 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Mussler,1, 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Schüffelgen,1, 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Grützmacher,1, 2 and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='03968v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The formation of a supercurrent through the Bi4Te3 layer is explained by a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' First, due to the close proximity of the Al/Ti electrodes a superconducting gap is induced within the Bi4Te3 layer right below the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The size of this gap is determined by analysing multiple Andreev reflections (MARs) identified within the devices differential resistance at low voltage biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The presence of these ballistic modes gives indications on the topological nature of Bi4Te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' INTRODUCTION Hybrid structures of three-dimensional topological insulators and superconductors are considered promising building blocks for the realization of topological quantum circuits [1– 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' A crucial optimization parameter is a sufficiently large coupling of the superconductor to the topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' By measuring the current-voltage characteristics of these junctions, the interface transparency as well as the underlying mode of transport, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' diffusive or ballistic, can be investigated [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The supercurrent in a Josephson junction is carried by electron-hole bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Based on the nature of these bound states their energy phase relation (EΦR) has a fixed periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Irradiating the junction with a radio frequency signal allows to investigate the Shapiro step response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' As the Josephson voltage V0 = hf/2e in ∗ rosenbach@ph2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='uni-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='de present address: Physics Institute II, University of Cologne, 50937 Köln, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In junctions with topological weak link both Andreev bound states (ABS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' diffusive bulk and surface modes) carrying 2e charge per cycle and Majorana bound states (MBS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' ballistic, perfectly transmitted surface modes) carrying only a single 1e charge per cycle coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Topological Josephson junctions can be separated into two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The first region is the topological matter underneath the superconducting electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Here, the proximity effect opens an effective induced superconducting gap within both the surface and bulk of the topological matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For the investigation of novel topological matter the question arises what relevant transport channels exist and what is their main mode of transport, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' ballistic topological surface states or diffusive bulk states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Initially, Bi4Te3 has been reported to be a zero band gap semimetal, comprising a Dirac cone at the Γ-point [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In advanced GW-band struc- ture calculations a band gap of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='2 eV was identified around the Γ-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Owing to time-reversal and mirror symmetries, Bi4Te3 is a strong topological insulator (STI) as well as a topological crystalline insulators (TCI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Furthermore, it is predicted that it contains higher order topological states [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' This approach allows to achieve a clean interface between the Bi4Te3 layer and the superconductor without any contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' EXPERIMENTAL SETUP Nanoribbon Josephson junctions have been defined following an all in situ approach[7, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Therefore, two independent masking techniques are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The first two layers are 5 nm of oxidized SiO2 and 15 nm of low pressure chemical vapor deposited (LPCVD) Si3N4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' They comprise the selective area growth (SAG) mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Afterwards, a 300-nm-thick SiO2 layer and a 100-nm-thick Si3N4 layer are deposited using LPCVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' These layers comprise the stencil mask used to deposit the superconducting electrodes in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The HF dip also locally removes the SiO2 of the first oxidized layer of the SAG mask only within the Si3N4 nanotrenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' During molecular beam epitaxy the Bi4Te3 will selectively grow within these nanotrenches on top of the Si(111) that is revealed during SiO2 removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The free-hanging Si3N4 bridge structures will be used after the deposition of Bi4Te3 to define the superconducting electrodes, without breaking the vacuum [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Bi4Te3 is a stoichiometric state of the (Bi2Te3)m(Bi2)n family with (m : n) = (3 : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 4 A unit cell comprises an alternating stacking sequence of Bi2Te3 quintuple layers and Bi bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In order to acquire Bi4Te3 nanostructures, the optimum growth parameters are subjected to the pre-patterned substrates with combinational surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The substrate rotation ensures a homogeneous growth of the Bi4Te3 layer also underneath the free-hanging Si3N4 bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The thickness of the Bi4Te3 nanoribbon depends on the geometry and width of the nanotrenches [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For the junctions investigated here their respective thicknesses are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The superconducting electrodes are deposited within a nitrogen cooled chamber below 0◦C by turning off the substrate heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In situ deposited Bi4Te3 nanoribbon Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' a) shows a false-colored SEM graph of the top view of an Al/Ti - Bi4Te3 - Ti/Al Josephson junction as prepared in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The cross section along the nanoribbon main axis is schematically depicted in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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 ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='6 176 120 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='12 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='5 500 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='52 2 500 130 10 35 310 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='85 95 159 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='36 3 100 140 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='5 30 744 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='32 115 75 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='24 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Overview of interface parameters of Josephson junctions with Bi4Te3 nanoribbon weak link and Al/Ti (30 nm/3 nm) superconducting contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Given are the geometric dimensions, the junction width w, the junction length/electrode separation length L and the mean thickness t of the nanoribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' After electrode deposition devices are covered by a 5 nm thin Al2O3 dielectric layer electron beam evaporated from a stoichiometric target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' A false-colored scanning electron micrograph of an as-prepared Josephson junction with aluminum superconducting contacts is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Aluminum has previously been reported to diffuse into (Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='06Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='94)2Te3 thin films [18], which increases the interfacial resistance of junctions within the superconducting regime of the Al electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The critical temperature of the superconducting Al/Ti composite electrodes is determined to be Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The magnitude of the superconducting pair parameter has been determined to measure ∆ = 145 µeV, following Bardeen-Cooper- Schrieffer theory [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The electrodes of the in situ defined nanoribbon Josephson junctions are wire bonded to a chip carrier in a quasi-four terminal contact configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The junctions are cooled to a base temperature of T = 23 mK using a dilution refrigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 6 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' EXPERIMENTAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Multiple Andreev reflections We have measured three junctions of different width w = 100, 500, and 1000 nm (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Both dV/dI(I) as well as V (I) as a function of a d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' current bias applied for the 500,nm wide junction (junction #2) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 2 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For an applied bias current below the critical current of Ic = 35 nA (see also Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I) a Josephson supercurrent estab- lishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The differential resistance dV/dI is zero below I < Ic and reaches a finite value as soon as the current bias exceeds I > Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The critical current is not hysteretic, whether the current bias is swept from positive to negative current biases or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I the critical current Ic, the normal state resistance RN values are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The corresponding IcRN product values are found to be in the range between 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='85 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='32 µV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='8 80 160 (μV) MAR ord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' -1 (1/n) n=1 2 3 4 57 Vn FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' IV -characteristics and differential resistance dV/dI of Josephson junction #2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' a) IV -characteristics and differential resistance as a function of the applied d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' bias current (dV/dI(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' A linear extrapolation from the IV-characteristics above 2∆∗ to V = 0 is shown (red dashed line) to extract the excess current Iexc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' b) Differential resistance as a function of the measured d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' potential drop across the Josephson junction (dV/dI(V )), showing signatures of multiple Andreev reflections (MARs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The inset shows the position (Vn) of the MARs plotted against the inverse of the MAR order number (1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The linear fit is forced through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' layer is a two step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 b)), which decays over a length scale given by the superconducting coherence length ξN within the Bi4Te3 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For the superconducting coherence length we have to consider two different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' When the distance between two elastic scattering events exceeds the superconducting coherence length, the transition is in the ’clean limit’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' A) and an elastic mean free path length of only le ≈ 4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Furthermore, the Hall bar data does not show any significant increase of the magnetoresistance, as expected from a Dirac semimetal [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Only beyond a current bias of about |I| > 740 nA the junctions resistance is mostly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' At this point the potential drop across the junction measures 2∆∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' the size of the proximity induced superconducting gap, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In order to quantify the size of the induced superconducting gap ∆∗ in the proximitized Bi4Te3 more precisely (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 b)) we analyzed multiple Andreev reflections (MARs) visible within the differential resistance dV/dI of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In junction #2 we observe MARs of the order n = 1, 2, 3, 4, 5, 6, 9, 11, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Missing signatures of intermediate order MARs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' n = 7, 8) has been observed before in BiSbTeSe2 nanoribbon Josephson junctions [22] but an explanation is missing until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The induced supercon- ducting gap measures ∆∗ = 95 µeV (for n = 1, as indicated by a blue dot within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 2 b) at T = 50 mK), which is smaller than the gap of the Al/Ti superconducting electrodes (∆ = 145 µeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Temperature dependency of Ic and MARs Figure 2 b) shows the differential resistance of junction #2 at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The signatures of MARs vanish above the critical temperature of the Al/Ti superconducting electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The temperature dependency of MARs of order n = 1 (blue dots), n = 2 (orange dots) and n = 3 (green dots) is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 2 b) and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Above formula is fitted to the ∆∗(T) data and a value for γB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='36 has been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The γB values of the other two junctions are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The value of γB indicates that there is an effective barrier present between the Al/Ti layer and Bi4Te3 despite the in situ fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' As a next step, the Josephson supercurrent between the the proximitized regions with the superconducting gap ∆∗ is analyzed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The supercurrent depends on the kind of transport, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' ballistic or diffusive, and on the transparency between the proximitized Bi4Te3 layers and the Bi4Te3 weak link (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The transparency of the interfaces of the lateral Josephson junction are analyzed in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='8 V (µV) T (K) 2Δ* Δ* 2/3Δ* FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Each trace is offset by 25 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' b) shows the temperature dependent position of 2∆∗ and MARs of order n = 2 and n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' c) shows the temperature dependent IV -characteristics from 50 mK to 500 mK, where each trace is offset by 5 µV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The red dashed line indicates the critical current of the junction Ic = 35 nA at T = 50 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The temperature dependent critical current, Ic(T), is shown in the inset with the fit indicated by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 2 a)) to V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' An analytical expression following Niebler, Cuniberti, and Novotny [24] is used to determine the junctions interfacial barrier strength Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='86 using the parameter α = eIexcRN/∆∗ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The barrier strength is related to the trans- parency via τ = 1/(1 + Z2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Similar to the observation from Kunakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [25] we find that the values for the interface transparency parameters τ and γB are interdependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' This effect can be at- tributed to a metallization effect the electrodes have on the surface states of the Bi4Te3 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' An additional method to determine the interface transparency is by quasi-classical analy- sis of the temperature dependent critical current [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Using a voltage criterion the critical current is extracted from the IV -characteristics at different temperatures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 2 d), with the inset showing Ic(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' We used a ballistic model fit (shown as black dashed line in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 1 b)) based on the Gor’kov equations with arbitrary junction length L [26] and barrier transparency D [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For the fit a value for the critical temperature of the gap within the Al/Ti electrodes Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='95 K and a Fermi velocity of vF = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='8 × 105 m/s of the Bi2Te3 surface layer are used [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The best fit results in an interface transparency of D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='6, which is in good agreement with the transparency τ determined using the excess current analysis described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' We also performed a quasi-classical fit using a diffusive model based on the Usadel equations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' However, within a physically reasonable range of values we did not get a decent fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Our analysis indicates that the supercurrent is carried by ballistic modes with increased superconducting coherence length rather than bulk modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The superconducting coherence length of these ballistic modes can be estimated within the clean junction limit to measure ξN = ¯hvF/2πkBTc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Within the range of frequencies applied 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='7 GHz ≤ frf ≤ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz we observe full integer Shapiro steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' At frequencies frf = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz, how- ever, additional sub-integer Shapiro steps have been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In junctions of high transparencies or very short ballistic junctions the CΦR is expected to be non-sinusodial [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Already for values of ξN/d > 3 the CΦR is skewed and the maximum current density lies above a value of φ > π [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Skewed, non-sinusoidal CΦRs can be decomposed into sinusoidal components of lower periodicity, which can explain the evolution of sub-integer Shapiro steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For ballistic modes of increased superconducting coherence length (ξN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='15 µm) this limit would need to be considered as the junction length of junction #2 (L = 130 nm) is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For the wider junction (junction #1, w = 1000 nm) as for the narrower junction (junction #3, w = 100 nm, see supplementary Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' B) the sub-integer Shapiro steps have been observed as well, confirming the presence of ballistic modes independent of the junction geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz, which is the cross over frequency considering the whole supercur- rent is carried only by MBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' As no missing Shapiro steps have been recorded it is assumed that less then one third of the supercurrent is carried by MBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz f = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='7 GHz 1/2 1/3 1/5 b) dV/dI (Ω) f = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='7 GHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Shapiro response of Josephson junction #2 at different radio-frequencies ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The differential resistance is displayed in between values of dV/dI = 0 (red) and dV/dI = 320 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The sub-integer steps are highlighted with their given fractions of the first integer Shapiro step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 13 3 2 1 0 1 2 3 4 2 0 2 4 9 8 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='000 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='000 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='000 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In order to establish robust proximitzed regions underneath the Al/Ti electrodes the presence of bulk carriers are probably beneficial, if not essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The Bi4Te3 has been identified to carry a large amount of bulk charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The proximitized regions of the Bi4Te3 are coupled by the unproximitized Bi4Te3 weak link giving rise to a Josephson supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Analysing the temperature dependency of the critical current we indeed identified the transport regime in these junctions to be mainly ballistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' From our Shapiro step measurements we came to the conclusion that the current-phase relationship is non-sinusoidal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' supporting our claim of ballistic modes in our junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' However, we did not find a 4π contribution in the Shapiro step measurements indicating the presence of Majorana zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' One reason might be that our lowest rf frequency of f ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='7 GHz was too high, so that we could not enter the regime where the 4π contributions are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For future material combinations in hybrid Josephson junctions including topological matter it is important to consider our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' This work was financially supported by the German Federal Ministry of Education and Research (BMBF) via the Quantum Futur project "MajoranaChips" (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 13N15264) within the funding 14 program Photonic Research Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Competing interests The author(s) declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [1] F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Ishibashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Tarucha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Buhmann, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Molenkamp, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 12, 137 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Schüffelgen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rosenbach, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Luysberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Mussler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Berenschot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Tas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Golubov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Brinkman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Schäpers, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Grützmacher, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 14, 825 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rosenbach, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Jalil, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Mussler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Neumann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Trellenkamp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Golubov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Brinkman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Grützmacher, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Schäpers, Science Advances 7, eabf1854 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Dominguez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Kashuba, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Bocquillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Wiedenmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Deacon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Klapwijk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Platero, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Molenkamp, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Trauzettel, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Hankiewicz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' B 95, 195430 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Aktürk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Aktürk, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Ciraci, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Uesugi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Takeguchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Kolobov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Tominaga, Nanoscale 9, 15115 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 15 [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Petroff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' B 105, L081409 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Nabok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Tas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Kennedy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Chávez-Garcia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Jalil, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Bennemann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Trellenkamp, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Lentz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Neumann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Lindström, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' de Graaf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Berenschot, N.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Panghotra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Raes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' de Souza Silva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Cools, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Keijers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Scheerder, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Moshchalkov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Van de Vondel, Communications Physics 3, 53 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' [33] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Raes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Tubsrinuan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Sreedhar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Guala, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Panghotra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Dausy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' de Souza Silva, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Van de Vondel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' B 102, 054507 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 17 SUPPLEMENTARY INFORMATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Magnetotransport Besides Josephson junctions with a Bi4Te3 weak-link we have fabricated nano Hall bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The fabricated device is shown as a false color scanning electron micrograph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 5 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The nanoribbons have a width of w = 100 nm and the spacing in between two side nanorib- bons is L = 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Nano Hall bar devices have been cooled down to T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='5 K in a variable temperature in- sert cryostate equipped with a superconducting magnet that can apply magnetic fields up to Bmax = 13 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The sample holder insert is equipped with an electromechanical stepper motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' From Hall measurements in an out-of-plane applied magnetic field the Hall slope AH = dRxy/dB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='58 Ω/T and subsequently the two- dimensional sheet carrier density n2D = (AHe)−1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='9 × 1014 cm−2 have been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='5% over the whole range of applied magnetic fields B ≤ |±13T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In a perpendicular applied magnetic field (Θ = 90◦, red curve) the mag- netoresistance does show a spectrum of universal conductance fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The temperature dependent magnetoresistance data in a perpendicular applied magnetic field for temperatures in between 2 K ≤ T ≤ 30 K is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 5 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The value for rms(δGxx) is constant up to a temperature of 9 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For higher temperatures the values follow a T −3/2 dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=') 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='04 T (K) rms(δGxx)(e2/h) a) b) c) d) T-3/2 1 μm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Magnetotransport data on Bi4Te3 Hall bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' a) Layout of the selectively grown Bi4Te3 nano Hall bar investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' b) Longitudinal magnetoresistance as a function of magnetic field for various tilt angles (Rxx(B, θ) of the devices main channel w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='t the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The orientation of the sample is schematically depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' d) Temperature dependency of the longitudinal magnetoresistance (Rxx(B, T)) for temperatures in between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='5 K≤ T ≤ 30 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The inset shows the temperature dependency of the root mean square of the conductance fluctuation amplitude rms(δGxx(T) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' All the junction parameters are given within the table 19 BST2318 Bi4Te3 100nm HB Rxx(Q) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='8 760 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='2 755 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='1 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='8 750 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='3 745 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 10 5 0 5 10 B (T)BST2318 Bi4Te3 100nm FFT Ampl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' (°), L=1900nm 12 10 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 8 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 frequency (1/B) FFT Amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='u) 6 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='10 0 20 40 60 80 100 Deg (°)Mag = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='58 K X 1 μm WD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='4 mm EHT = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='75 GHz f = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 GHz f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='1 GHz 600 450 300 150 0 600 450 300 150 0 600 450 300 150 0 4 0 6 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Shapiro response of Josephson junction #3 at different radio-frequencies ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='75 GHz, f = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='25 GHz, f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 GHz and f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='1 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' The differential resistance is displayed in between values of dV/dI = 0 (red) and dV/dI = 600 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='75 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' in the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Next to these standard junction characeristics we here show Shapiro step measurements of the narrow junction #3, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' Within a similar range of radiofrequencies applied, as for junction #2 in the main text, we observe a similar behavior w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' the Shapiro step evolution in the differential resistance as a function of the applied RF power applied to and the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' potential bias applied across the junction (dV/dI(P, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' For the largest frequency applied of f = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='75 GHz, not only full integer Shapiro steps, but also half-integer Shapiro steps can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='000 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 160.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='00 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content='0would in topological insulator nanoribbons, where a quantization of transverse momentum states would alter the surface state dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQfjgcV/content/2301.03968v1.pdf'} diff --git a/3tE2T4oBgHgl3EQfOAZw/content/tmp_files/2301.03743v1.pdf.txt b/3tE2T4oBgHgl3EQfOAZw/content/tmp_files/2301.03743v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f251610ed40150059f4668929a8dee09d7e810d --- /dev/null +++ b/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. +3The +ZTF +data +can +be +obtained +from +IRSA + +using +the +inter- +face + +or +using +a +wrapper +of +the +above +IRSA +API +. + +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. + +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 +References +We provide two forms of the references section (for ADS and as published) so that the references can be easily +incorporated into ADS. +References (for ADS) +Collazzi, A. C., Schaefer, B. E., Xiao, L., Pagnotta, A., Kroll, P., Löchel, K., & Henden, A. A. 2009, AJ, 138, +1846 (arXiv:0909.4289) +Downes, R. A., & Shara, M. M. 1993, PASP, 105, 127 (https://doi.org/10.1086/133139) +Duerbeck, H. W. 1988, A&A, 197, 148 +Duerbeck, H. W. 1987, Space Sci. Rev., 45, 1 (https://doi.org/10.1007/BF00187826) +Evans, A., Gehrz, R. D., Woodward, C. E., & Helton, L. 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AJ 78, 375 + diff --git a/3tE2T4oBgHgl3EQfOAZw/content/tmp_files/load_file.txt b/3tE2T4oBgHgl3EQfOAZw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49b57e6d20f9c7cbbd8a231460cfb577fd8b238c --- /dev/null +++ b/3tE2T4oBgHgl3EQfOAZw/content/tmp_files/load_file.txt @@ -0,0 +1,607 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf,len=606 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='03743v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='SR] 10 Jan 2023 1 LS And: WZ Sge-type outburst first time since the 1971 discovery Taichi Kato1 tkato@kusastro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Using the Zwicky Transient Facility (ZTF) data, I found that the object underwent the second known outburst in 2022 April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The true observed peak of the 1971 outburst was likely 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The outburst parameters were similar to those of other WZ Sge-type dwarf novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The fading tail lasts more than a year and the object is still currently on this tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' There was a hint of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='5-mag temporary brightening on the fading tail and the object appears still active after the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' LS And was discovered by van den Bergh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' (1973) in the region of M 31 (named “m” in their paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' van den Bergh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' (1973) stated that the object was visible only on a blue and on a yellow plate taken in immediate succession on 1971 August 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' van den Bergh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' (1973) suggested that the variable might be either a supernova or a flare star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Although van den Bergh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' (1973) did not give the brightness of this object, it was estimated to be 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='5 from their figure by Romano (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Sharov (1973) examined plates taken in the Crimean Station of Sternberg Astronomical Institute and Latvian Radio Astrophysical Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Sharov (1973) succeeded in obtaining one observation near the maximum and the light curve of the fading part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Sharov (1973) noted the presence of a star of 21–22 mag on Palomar Observatory Sky Survey (POSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Based on the large amplitude exceeding 8 mag, rapid fading (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='2 mag d−1) in the early fading part and the very slow (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' intergalactic nova) if it was indeed a nova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Romano (1977) examined Asiago plates and presented a rough light curve of the outburst (probably unaware of the work by Sharov 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Romano (1977) indicated that the variable was at the limit of visibility (∼20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='5 mag) on POSS and that color was almost white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Romano (1977) concluded that this object is probably a dwarf nova of UV Per type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='1 Following Romano (1977), Meinunger (1977) studied Sonneberg plates (probably also unaware of the work by Sharov 1973) and constructed a light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Meinunger (1977) suggested that this object was probably a very bright nova in the halo of M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Precise astrometry of the outbursting object using the materials at Latvian Radio Astrophysical Observatory indicated the identity with the object on POSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' In GCVS version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='2 for extragalatic variables, LS And was also given a name M31V0002 probably reflecting the possibility of an object in M 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 1UV Per was considered to be the prototype of dwarf novae with large-amplitude and rare outbursts at that time (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Petit 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' WZ Sge was considered as a recurrent nova and the concept of WZ Sge-type dwarf novae was not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' See Kato (2015) for a modern review of WZ Sge-type dwarf novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2 Table 1: Observations of the 1971 outburst of LS And.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' JD∗ mag† source‡ JD∗ mag† source‡ JD∗ mag† source‡ 179 [19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='0 3 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='497 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='30 2 292 [19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='0 3 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='468 [20.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' † [ upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' : uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' * eye estimate from the published figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' ‡ 1: van den Bergh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' (1973), 2: Sharov (1973), 3: Romano (1977), 4: Meinunger (1977), 5: Sharov and Karimova (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The sources are vdB73 (van den Bergh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 1973), S73 (Sharov 1973), R77 (Romano 1977), M77 (Meinunger 1977) and S78 (Sharov and Karimova 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The “v” symbols represent upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Although most professional astronomers considered or treated LS And as a nova (Downes and Shara 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Szkody 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Collazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Özdönmez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2018), and some suspected to be an X- ray nova (Rosenbush 1999) or a recurrent nova (Duerbeck 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' A part of the atmosphere in the late 1980s among amateur astronomers was already told in Kato (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Visual monitoring of LS And for a new outburst already started in 1987 by VSOLJ members, and then by observers around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2001, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The blue color (Gaia B − R=+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='25) and a large proper motion were, however, sufficient to convince me of the dwarf nova-type nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The parallax in Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2021) was not conclusive, probably due to the faintness of this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The color in Gaia EDR3 was even bluer (B − R=−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' The “moment” arrived like lightening when I was examining light curves obtained by the Zwicky Transient Facility (ZTF: Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 2019)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' It was when I started examining light curves of recent ZTF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' 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”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' Unthinkable!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE2T4oBgHgl3EQfOAZw/content/2301.03743v1.pdf'} +page_content=' I initially could not 2 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 + diff --git a/79AyT4oBgHgl3EQfQvZ4/content/tmp_files/load_file.txt b/79AyT4oBgHgl3EQfQvZ4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..859e0c9777cb1359b3c72f0cbc1399bc91bd6281 --- /dev/null +++ b/79AyT4oBgHgl3EQfQvZ4/content/tmp_files/load_file.txt @@ -0,0 +1,317 @@ +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': 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0000000000000000000000000000000000000000..53c2cf8e1b0aba985484b09fef8bb7918a696935 --- /dev/null +++ b/89FLT4oBgHgl3EQfti_6/content/tmp_files/2301.12152v1.pdf.txt @@ -0,0 +1,1524 @@ +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. + + + + + +<div> +<p> +<img> +<html> +<head> +<meta name="viewport" content="width=device- +width,initial-scale=1"> +<link href=" style.css" rel=" stylesheet”> +<title>Critical Path + + +

Hello web performance +students!

+
+ + +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 tag is usu- +ally the illustration of its adjacent (e.g., parent) node with +
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. 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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. 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Bioinformatics 37 (2021), 360 – 366. + +安全国品电子电器政装用品 +外饰用品内饰用品养护用品自空游用品正报价 +M +Opma适8轮胎防摄膜章道询 +昆聘特区：车博土 +巴洛克 +喜普风 +香玉儿 +迪佳 +键科 +暖忆 +H +H +H同酮片购买 +Q@: +你可自能摄线： +吃米非司酮片后流血 +吃完米非司酮片的反应 +吃了米非司酮片流血 +吃米非司酮片有什么症状 +服米非司酮片 +吃米非司酮片第二天出血 +吃米非司酮片的注意事项 +吃米非司酮片会流血吗 +开门红 +咨询药师 +药师微信什公值得买 +Q搜索分类/品牌/商品 +打开 +全部奶价 +社区 +商品百料 +抗事等 +要爽玩《魔兽世界：争霸艾泽拉斯》 +CPU +鲁118-88-50 ++美注 +RYZEN +WORLDL +于一 +量 +家宝业，额心后上靠玩安成的原国，可情不远 +大本身视普世养料 +MAGB550MMORTARWiFI迫击炮+... +￥2392 +1899 +抢 +市场价 +值得买APP专享价代公值得买 +Q搜索分类/品牌/商品 +打开 +泡好价 +全部好价 +社区 +商品百料 +捷惠等 +甲全部评论(119) +13心 +3068-08-0 +心 +ALARE +心 +3008488-30 +爱有年天 +4心 +MAGB550MMORTARWiFI迫击炮+... +X +¥2399 +1899 +抢 +市场价 +值得买APP专享价S搜狐网 +晴/1°三 +如何搭迪和自己不熟的女同事-如何搭让 +如何搭训技巧之一：微笑地说出对方的名字 +对于安生而言，如果一入男士非常绅士地对 +她微笑，并具当着她的面，自然友好地叫出 +了她的名字，她肯定会感到惊访，但随之而 +来的更多是欢喜。这种搭训会让女生瞬时记 +住自己，并且留下较好的印象。 +因为你微笑对她，她也会回以礼貌的微笑 +然后她会反问：“你怎么知道我的名学”。就 +解中度过愉悦的时光。彼此都给对方以缸服 +的感觉，这为下次的聊天或相聚做好 +垫。 +讨论5女相100男，徐州一相亲大会男女比例失调..> +男 +女 +我来说两句 +C知乎 +Q中国灵.. +下载App +注册登录 +不认识的同事（女）如何搭训？ +关注问题 +写回答 +1个回答 +小白兔 +谢邀，建议先从她认识的人入手，比如拿联系方 +式，然后可以先切入聊天 +发布于2020-11-1421:17 +一费同 +评论 +智能消费 +新浪潮 +创卷调研 +广告 +相关推荐 +男朋友打游戏正确的处理方式，这个 +女朋友不能要了 +微博的广告 +如何去搭陌生人 +haoyunlai2188的文章 +ApP内打开Layout-aware Webpage Quality Assessment +SIGKDD ’23, August 06–10, 2023, Long Beach, CA +[3] Yoshua Bengio, Aaron Courville, and Pascal Vincent. 2013. +Representation +learning: A review and new perspectives. 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In Proceedings of the 27th ACM SIGKDD +Conference on Knowledge Discovery & Data Mining. 4014–4022. + diff --git a/89FLT4oBgHgl3EQfti_6/content/tmp_files/load_file.txt b/89FLT4oBgHgl3EQfti_6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec3fa0ad688fc8e572951d01d75021986618bdeb --- /dev/null +++ b/89FLT4oBgHgl3EQfti_6/content/tmp_files/load_file.txt @@ -0,0 +1,1007 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf,len=1006 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', Beijing, China {chenganfeng01,liweibin02,wangshuaiqiang,huangzhengjie,fengshikun01,chengzhicong01}@baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='com liuyiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='tanh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='com,dq22@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='cn,yindawei@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='org ABSTRACT Identifying high-quality webpages is fundamental for real-world search engines, which can fulfil users’ information need with the less cognitive burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Early studies of webpage quality assessment usually design hand-crafted features that may only work on par- ticular categories of webpages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', shopping websites, medical websites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' They can hardly be applied to real-world search engines that serve trillions of webpages with various types and purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In this paper, we propose a novel layout-aware webpage quality assessment model currently deployed in our search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Intu- itively, layout is a universal and critical dimension for the quality assessment of different categories of webpages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Based on this, we directly employ the meta-data that describes a webpage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', Doc- ument Object Model (DOM) tree, as the input of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The DOM tree data unifies the representation of webpages with different categories and purposes and indicates the layout of webpages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Moreover, we improve the GNN method with an attentive readout function, external web categories and a category- aware sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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§.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Layout- aware Webpage Quality Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In SIGKDD ’23: ACM Special Interest Group on Knowledge Discovery and Data Mining, August 06-10, 2023, Long Beach, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' ACM, New York, NY, USA, 11 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' XXXXXXX ∗ Co-first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' § Dawei Yin is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' SIGKDD ’23, August 06–10, 2023, Long Beach, CA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' ACM ISBN 978-1-4503-XXXX-X/18/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION Search engines, such as Google and Baidu, plays an important role in fulfilling users’ information need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' For example, given webpages with comparable relevance, high-quality webpages should be ranked higher than its competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' How- ever, modern search engines usually face trillions of webpages with various categories, where simple hand-crafted features and classification algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Bayesian Networks [7]) can hardly capture the in-depth information that reveals the webpage quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Moreover, most of them can only work on a particular category of webpages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', shopping websites [8], medical website [25], web portals [6], and Wikipedia articles [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' They are hard to be effec- tively applied to real-world search engines that serve trillions of heterogeneous webpages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To address the aforementioned limitations, we conduct the first work that investigates layout-aware webpage quality assessment on real-world web data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Modeling in-depth layout information is promising for webpage quality assessment in real- world web search scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='12152v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='IR] 28 Jan 2023 SIGKDD ’23, August 06–10, 2023, Long Beach, CA Cheng and Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' <div> <p> <img> <html> <head> <meta name="viewport" content="width=device- width,initial-scale=1"> <link href=" style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='css" rel=" stylesheet”> <title>Critical Path

Hello web performance students!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='

Figure 1: A toy example that shows the webpage layout represented by the DOM tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' RQ2: How to encode webpage layout information for webpage quality assessment on large-scale heterogeneous webpages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To answer RQ1, we propose to extract layout information of webpages from Document Object Model (DOM) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Figure 1 shows a toy example of the hierarchical structure of a DOM tree, which is converted from its HTML source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', type and size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Such data indicates rich hierarchical in- formation on the content and its layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' And different categories of webpages can be represented in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' It is indisputable that inspecting the structure of DOM tree can help to measure the quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Notably, Graph Neural Networks [12, 15] has shown great per- formance in modeling structured text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', word interactions) [18, 37, 41], yet they are unexplored for complex DOM tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Additionally, we deployed our model in the online ranking system and achieve good improve- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Last but not least, the proposed solution is currently fully deployed in the online system of Baidu Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Overall, our main contributions can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We conduct rigorous offline and online experiments before fully deploying the model online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The experimental results show that the proposed solution is effective to be applied in real-world search engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1 Graph Neural Network Recent years have witnessed the success of graph neural networks (GNNs) for relational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GraphSAGE [15] is proposed to generalize GCN’s aggregation operation from average to sum, max and a RNN unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' For more details, please refer to [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' For example, Sanyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GraphQA [2] is a graph-based method to estimate the quality of protein models, that possesses favorable properties such as representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Schweidtmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [27] develop GNN models for predicting three fuel ignition quality indicators of oxygenated and non-oxygenated hydrocarbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [36] propose WTAGRAPH, a web tracking and advertising detection framework based on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Notably, a handful of researches [18, 37, 41] leverage GNN to perceive text structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', word interactions) for down-stream tasks, which are the closest research to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' However, they mainly consider the relationships between segments of text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', words and paragraphs), and do not consider the overall structure and layout of webpage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', how the multi-modal content is orga- nized and presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In addition, the expressiveness of GNN is not explored for the task of webpage quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Layout-aware Webpage Quality Assessment SIGKDD ’23, August 06–10, 2023, Long Beach, CA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2 Layout Layout, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Substantial efforts have been made to explore the importance of layout in many AI areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' For example, Besides, Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [4] design an automated deep generative model using graph neural networks to generate synthetic data with highly variable and plausible document layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Avetisyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [40] propose a new kind of classification method for lithography layout patterns based on graph convolution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Recently, with the development of pre-trained language models, LayoutLM-style methods have achieved success in textual or multi- model document understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' LayoutLM-v3 [20] pre- trains multi-modal Transformers for Document AI with unified text and image masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Our work differs from the aforementioned studies, as our main focus is to model the layout of webpages for quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 3 PRELIMINARIES In this section, we introduce the basic concepts and formalize the problem of webpage quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We summarize the com- monly used notations in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1 Layout-aware Webpage Quality Intuitively, high-quality webpages are those that clearly provide useful information for users in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Specifically, given a set of webpages with comparable relevance under the same query, we consider the layout (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' structure design, content presentation) as the key dimension of measuring webpage quality and improving user experience [6, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We give the definition Table 1: Commonly-used notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Notations Descriptions G𝑝 = {N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' E} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='A layout graph of webpage 𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The node set of graph G𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The edge set of graph G𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='F = {F𝑡 }𝑇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑡=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The layout-related feature sets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The number of node type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='F𝑡 = {f𝑖}|F𝑡 | ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The feature set of node type 𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='f𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The layout-related feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='E(·) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The embedding of its input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='�ℎ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The initialized embedding of node 𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑡𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The node type of node 𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝜷𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The category of webpage 𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='�ℎ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑣 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The initialized embedding of virtual node 𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝜎(·) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='An activation function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝛼𝑛𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The attention score between nodes 𝑚 and 𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑒𝑛𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The attention coefficient between nodes 𝑚 and 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑠𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The predicted assessment score of webpage 𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑦𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='The manually assessment score of webpage 𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='and some examples for each aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The rules and principles for annotators are defined as the following: 0 means poor layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' On the basis of ordinary pages, points will be deducted for various flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 1 means ordinary layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 1 point is common, and annotators are required to add or deduct on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='5 means better structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' a certain gain compared to ordi- nary layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 2 means gainful layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The user experience of this layout is significantly better than most layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Finally, annotators are required to score the give webpage from 0 to 2 points based on the above rules and principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2 Layout Graph To extract quality information from webpage layout, we construct a layout graph for each webpage based on its DOM tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In particular, a layout graph is denoted as G𝑝 = {N, E} that contains a node set N and an edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Each node has a specific type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', text, image and video), and is associated with several layout-related features F𝑡 = {f𝑖}|F𝑡 | 𝑖=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The features of different types of nodes are denoted as F = {F𝑡 }𝑇 𝑡=1, where𝑇 is the total number of node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Table 2: The considering aspects of rules and principles to score the layout of a webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The detailed construction process of layout graph is depicted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' its quality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e, 𝑠𝑝 = 𝑓𝜃 (G𝑝, F𝑡, 𝜷𝑝), (1) where 𝑓 (·) represents the quality model, and 𝜃 denotes its param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The scores should be consistent with users’ perception of webpage quality, and reflect the rules and principles as we described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 4 METHOD In this section, we first present the overview of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Then, we describe the graph formulation process for a webpage, including the construction of the layout graph and feature pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' After that, we present a GNN-based solution for webpage quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1 Overview Our solution mainly contains two components: layout graph for- mulation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2) and quality assessment model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', 𝜷𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Next, we propose a quality assessment model that leverages Graph Attention Network (GAT) to perform expressive message passing between nodes in the layout graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Both local and global structure information of the layout graph can be encoded in latent representations, which are exploited for the quality assessment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2 Layout Graph Formulation Graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The content layout has been viewed as one of the most critical dimensions for measuring webpage qual- ity [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To formulate layout information for various categories of webpages, we first construct layout graph based on DOM tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In Table 3: The summary of selected features for each node type in our constructed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Then, Depth First Search (DFS) is used for exact- ing adjacency relationships from the DOM tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Specifically, we recursively record the nodes and the corresponding edges between parent and child nodes in the DOM tree, as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The layout graph G𝑝 of webpage 𝑝 can be expressed by the exacted nodes N and their relations E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Virtual node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' As such, we can capture global information of the given graph via the virtual node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Feature pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To capture the layout information of the webpage, we design a series of features for each node type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Taking the text node as an example, font style, font size, alignment and position in webpage are all represented by learnable embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The detailed list of features is presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' More specifically, for continuous features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The uniform division of the whole interval leads to the data sparse issue since the continuous features typically obey a long-tail dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Discrete features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 1We parse webpages with the python library: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='crummy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='com/software/ BeautifulSoup/bs4/doc/ Layout-aware Webpage Quality Assessment SIGKDD ’23, August 06–10, 2023, Long Beach, CA Layout-Related Features .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Virtual Node Layout Graph Node Features Graph Features Projection Page Score Figure 2: The illustration of message passing in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The red node represents the virtual node in the constructed layout graph, which is utilized for capture the graph-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GraphConstruction(root) begin N𝑟 = {𝑣𝑖𝑟𝑡𝑢𝑎𝑙_𝑛𝑜𝑑𝑒, root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛𝑜𝑑𝑒};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' E𝑟 = {(𝑣𝑖𝑟𝑡𝑢𝑎𝑙_𝑛𝑜𝑑𝑒, root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛𝑜𝑑𝑒)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' for child ∈ root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑐ℎ𝑖𝑙𝑑𝑟𝑒𝑛 do N𝑐 = child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛𝑜𝑑𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' N𝑟 = N𝑟 ∪ N𝑐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' E𝑐 = {(𝑣𝑖𝑟𝑡𝑢𝑎𝑙_𝑛𝑜𝑑𝑒, child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛𝑜𝑑𝑒), (child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛𝑜𝑑𝑒, root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='𝑛𝑜𝑑𝑒)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' E𝑟 = E𝑟 ∪ E𝑐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' N𝑐, E𝑐 = GraphConstruction(child);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' end return N𝑟, E𝑟 end G𝑝 = {GraphConstruction(Rp)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In addition to the node-level features, graph-level feature em- bedding is introduced to the layout graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', webpage category) to provide the model with the ability to perceive different cate- gories of webpages, which is vital to the quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' One reason is that the same webpage category has a similar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Therefore, with this graph-level embedding, the predicted assessment score shall be more robust in the online search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Another reason lies in that different webpage categories have different criteria for quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In summary, it is meaningful and important to take the graph-level feature embedding into account for the layout graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In particular, we consider two types of relationships in the graph that could be discriminative for the task: Local relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Intuitively, the relationships between adjacent nodes in the layout graph are important to reveal content quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' For example, a node with tag is usu- ally the illustration of its adjacent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', parent) node with
tag, which contains textual description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Global relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Another important insight is that the relationships between local content and global layout should also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Attentive message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To achieve this, we leverage graph neural networks that are promising to capture such complicated patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Here, �ℎ(𝑘) 𝑛 represents the embedding of node 𝑛 in the 𝑘-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Formally, the attention value SIGKDD ’23, August 06–10, 2023, Long Beach, CA Cheng and Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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 ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Note that the weight matrices are shared across different nodes in a single graph attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' After 𝐾 times of message passing, the layout-aware patterns could be captured by node interactions (as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (2)) within 𝐾-hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' It is worth noting that the virtual node also plays an im- portant role during the message passing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Overall, the GAT-based message passing framework is able to comprehensively model both local and global relationships for the final task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Readout function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (5) as 𝑠𝑝 = W�ℎ(𝐾) 𝑣 + 𝑏, (6) where �ℎ(𝐾) 𝑣 is the virtual node representation in 𝐾-th layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' the last layer) of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Category-aware quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The quality score defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (6) is based on rich information aggregated from nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' How- ever, graph-level information is critical yet not incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' There- fore, we further improve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (6) with the category information of webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In particular, we denote the category embedding of a given webpage 𝑝 as E(𝜷𝑝), and further rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (6) as 𝑠𝑝 = W(�ℎ(𝐾) 𝑣 + E(𝜷𝑝)) + 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Category-aware data sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' As the graph-level category embedding is introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (7) to perceive different categories of webpages, the bias in different categories may affect the predic- tion of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In particular, some webpages are highly similar in layout, such as some popular question-answering websites, which are generated from templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Such webpages typically have sim- ilar layout scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Consequently, the predicted assessment score may be dominated by the category-aware embedding (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' graph level embedding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' To alleviate this issue, a category-aware sampling strategy is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Optimization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' After up-sampling, the model could be optimized through Mean Squared Error (MSE) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 5 DEPLOYMENT In this section, we show how the layout-aware webpage quality assessment model be applied to our online ranking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The overview of deployment is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1 Offline Input Data Construction In the left component of Figure 3, we present the process of input data construction for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Firstly, each webpage on the world wide web will be parsed through our HTML parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' All features of the HTML are stored in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Note that this process runs offline, it can significantly reduce the computing time of the online search system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We also list the features which are used in our webpage quality assessment model, details are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We classify the features into three main categories w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', location, content, and layout according to the different roles they play in building webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Category location is the primarily feature that locates the position of elements in the webpage e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', height, width and position type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Category content contains text-related features e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', the number of words, font style, and line height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Category layout is a feature that controls the layout of elements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', border, padding, and margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2 Online System Workflow The online system workflow is presented in the right component of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Our ranking system contains a wide variety of webpage features, where quality is one of the most important factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' height width paddling margin font border .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='. Layout-aware Quality Assessment Model Quality Score Database Online Ranking System new feature ranking list of each webpage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='. Ranking System new ranking results Input Data Construction Online System Workflow Figure 3: The overview of deployment in online ranking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' which is shown in the lower left area of the online component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In our experiments, we use 600,000 webpages for training and 20,000 webpages for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2 Evaluation Metrics Positive-Negative Ratio (PNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We use PNR to measure the con- sistency between manual quality labels and the scores estimated by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In particular, by enumerating all the pairs of webpages in the dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', I (𝑎 > 𝑏) = 1, if 𝑎 > 𝑏, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Here, 𝑓 (𝑑𝑖) represents the quality score of a webpage 𝑑𝑖 estimated by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Higher PNR value indicates better perfor- mance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Area Under Curve, Precision, Recall, F1-Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We also report Area Under Curve (AUC), Precision (P), Recall (R) and F1-Score (F1) to evaluate our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Precision and recall are often in tension, that is, improving precision typically reduces recall and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' F1-Score combines them to one performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Area under curve summarizes the trade-off between the true positive rate and false positive rate for a predictive model using different probability thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GIN [33] introduces a learnable parameter to adjust the weight of the central node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GAT [30] lever- ages the attention mechanism to improve neighbor aggregation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Our proposed models: Virt-GIN has a more expressive readout mechanism by adding the virtual node �ℎ𝑣 to GIN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Virt-GAT is our approach similar to virt-GIN model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', a GAT model with virtual node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Models-NC: Note that all the above- mentioned models use category information as proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' This can clearly illustrate the improvement brought by the proposed solution for our search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='4 Experimental Settings In our experiments, Adam is selected as the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We use the following hyper-parameters: embedding size (64), number layers (5), dropout probability (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2), batch size (32), learning rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='0001) for GNN models, train epochs (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' As for the TreeLSTM model, we set the embedding size (64), dropout probability (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='5), batch size (128), learning rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='0001), epochs (25) for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We run 5 experiments with different random seeds for all models mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The <广告> iMac 新开篇 进一步了解》 购买> AirTag 丢三落四这门绝技，要失传了。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 进一步了解> 购买>米非可酮片购买 QQ: 你可能还想找： 吃家非司限片有什么及应 吃家非司限片有什么别作用 吃家非司职片会出自码 第一天吃来非可期片 晚来非脲片 晚完来非司片的反度 晚来非司限片有什么用 惊 咨询药师 吧药师微信用品益新技检影音乐/安全用品/电子电型改装用 品/外维用品内信用品/养护用品自然范用品工投 检 品牌特区：车墙土 送进佳 Z室组调，今天小学生网小编竭据老师给大家整理了 关于汉字（室》的绳调列表，基望下西整理的竞字 组调资科及调语解择内容能够助到大家， 室字简介 首字母：y，群膏：yu，等声调拼音：yo，注音： U，部首：穴，部首比划：5，比划：15，第体 字：毫，字体结构：上下结构，第画顺序：擦擦折 PWRY，五第98编码: PWRY， Unicode : 服擦操推所除择服摄折探除，五笔86编码 U+7AB3，双字编号：6008, 基本解释 宝yo Uo（事物）思务，租务：室务，室 败（房效；' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='数坏），室陷（雅务，质量根差），良 室（优务），0量：室情，0蜜第 京组调 掌室(beny）：掌重相劣，清线源（圣式记》 卷/：“面官修战股，零意不能放洋，转座高组力 剩摄之用。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 事室(bbye)：(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='泄气；' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='干事，如：气球欢得个 头抵大，但用针一别就癌富了，(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='坑，童度，杨 《麦子黄时》：“自卫队上操，有时练习石 锁，他能单手掌置负子，一口气连孕十几下，后一 敬手，稳出七八步运，肥场地打个大靠，SIGKDD ’23, August 06–10, 2023, Long Beach, CA Cheng and Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Table 4: Offline experimental results of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Model PNR AUC (%) label 0 label 1 P (%) R (%) F1 (%) P (%) R (%) F1 (%) Online Baseline 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='51 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='10 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='44 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='99 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='39 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='09 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='57 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='60 TreeLSTM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='01 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='07 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='01 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='07 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='03 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='06 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='06 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='05 GIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='05 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='20 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='39 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='04 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='17 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='44 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='96 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='64 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='45 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='26 GAT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='06 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='23 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='87 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='03 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='93 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='21 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='69 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='52 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='65 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='40 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='75 Our Approach Virt-GIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='03 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='10 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='41 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='49 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='61 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='20 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='56 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='14 Virt-GAT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='10 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='24 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='57 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='17 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='13 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='35 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='79 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='24 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='71 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='29 Non-Category (-NC) GIN-NC 4,15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='07 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='30 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='36 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='42 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='29 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='57 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='25 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='19 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='24 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='62 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='20 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='15 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='17 GAT-NC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='04 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='15 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='23 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='75 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='31 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='67 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='87 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='25 Virt-GIN-NC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='04 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='14 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='53 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='49 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='27 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='44 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='86 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='77 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='33 Virt-GAT-NC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='03 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='08 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='48 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='40 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='23 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='43 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='04 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='86 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='60 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='25 final result we reported is the mean test AUC, Precision, Recall, F1-Score and their corresponding standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' All the above mentioned GNN models are implemented by Paddle Graph Learning (PGL)1, an efficient and flexible graph learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='5 Offline Experimental Results We report the offline experimental results of the proposed model and all baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Besides, we also include a baseline method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', the model that is used in the system before deploying the layout- aware webpage quality assessment model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', Δ𝐴𝑈𝐶 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='08, Δ𝐹1 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='96 (label0) and Δ𝐹1 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='97 (label1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Especially for PNR, where the value is improved from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='51 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' These tell us that the proposed model prefers high-quality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' By applying the proposed readout function, the model can have a significant improvement on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Especially, the new readout mechanism is able to improve PNR by a margin of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='38 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='96 based on GIN and GAT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='59%, Δ(𝑉𝑖𝑟𝑡_𝐺𝐼𝑁,𝐺𝐼𝑁 ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='22%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' All these phenomena show that our readout mechanism is capa- ble of improving the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Comparing the results of the two models whether apply the category-aware optimization strategy (w,r,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', GIN-NC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GIN, Virt-GIN-NC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Virt-GIN, GAT-NC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' GAT, Virt- GAT-NC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Although a few models obtain lower 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='com/PaddlePaddle/PGL values on a few metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', the F1-score of Virt-GAT-NC on label0 is 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='62 while Virt-GAT is 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='35, the precision of Virt- GAT-NC is 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='74% but Virt-GAT is 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='75%), the models with category-aware optimization show more robust performance considering all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='31, Δ𝐴𝑈𝐶 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' For high-quality webpage (label1) Δ𝑅 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='67%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' These large margins suggest that our model is more expressive than TreeLSTM, although TreeLSTM is specifically designed for tree-structured network topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='6 Varying the number of GNN layer In general, a webpage is represented as a DOM tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Its depth deter- mines how many layers of GNN are needed to obtain information from the root node to the leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' However, as the number of GNN layers increases, the computational efficiency will be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Therefore, we provide an experiment to verify the influence of the number of layers on the experimental results, as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' As seen from the table, the more layers, the higher the AUC score can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' However, compared with the 5-layer virt- GAT model, the improvement of 7-layer virt-GAT model is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' #Layers AUC (%) label 0 label 1 P (%) R (%) F1 (%) P (%) R (%) F1 (%) 1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='23 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='92 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='46 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='49 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='85 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='82 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='72 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='42 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='66 3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='27 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='64 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='80 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='06 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='86 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='79 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='05 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='16 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='45 5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='24 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='57 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='17 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='13 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='35 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='79 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='24 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='71 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='29 7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='22 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='80 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='77 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='61 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='41 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='32 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='43 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='42 we conduct a manual evaluation on the final ranking results with some real user-generated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' This directly reflects the quality of the results exposed to the end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We log a set of (million-scale) online queries and the correspond- ing final impressions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Note that the data logging is conducted by multiple rounds to eliminate randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We filter out examples in which queries have identical impressions between the two systems, and then utilize the rest for the manual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1 Online Experimental Metrics As mentioned in Section 5, our proposed quality assessment model works in Baidu retrieval system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The online experiments major focus on the end-to-end evaluation, the metrics are often used to measure the effectiveness of information retrieval system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Details are as follows: Discounted Cumulative Gain (DCG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We first log a dataset and manually label the data with 0 to 4 grades, and then report the relative improvement w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' the average DCG over the top-4 final results of all queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Additionally, we also report the relative improvement of DCG for the low quality ranking result w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=', manually label is 0/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Side-by-side Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Besides, we also conduct a side-by- side comparison between the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Here, the relative gain is measured Good vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Same vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Table 6: Discounted cumulative gain on manual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Rand-Query Tail-Query Same-Quality Δ𝐷𝐶𝐺 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='19% +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='42% DCG_0/1 ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='63% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='56% Table 7: Side-by-side comparison on manual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Rand-Query Tail-Query Same-Quality Δ𝐺𝑆𝐵 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='10% +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='52% +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Especially for DCG_0/1 ratio, the relative improvement values are respectively −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='63%, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='56% for rand query and tail query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The conventional case-by-case comparison also has signifi- cant improvement over the base system, especially for the rand query (Δ𝐺𝑆𝐵 = +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' This tells us that user experi- ence can be improved by taking into account the web page quality in search system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In addition, we can observe that with comparable relevance, the GSB value of the quality improvement is Δ𝐺𝑆𝐵 = +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='13%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' This intuitively shows that our new system can provide higher quality search results based on the guaranteed rele- vance of search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Moreover, we perform the statistical test to estimate whether the experimental results is statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The p-value of DCG rand and tail query are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='0613 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1276, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' (a) Offline quality assessment (b) Online position changes Figure 4: The overview of case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' system, which can demonstrate that our experimental results are statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Overall, the online experimental results show that our proposed layout-aware quality assessment model can effectively improve the performance of real-world ranking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' These typically cases are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1 Offline Quality Assessment In Figure 4(a), we present three webpages with different layout styles and their quality assessment scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The first webpage has a chaotic layout, elements in this web- page are unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' It affects the user’s normal browsing and is very difficult for user to obtain information from this webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Our quality assessment model marks this webpage as low quality (𝑠𝑐𝑜𝑟𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='0068).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' This extremely low score will be considered by the ranking system to lower its ranking position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The second webpage also has low quality, different with the chaotic layout of the first webpage, it has a normal layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' The ranking system can judge this by our quality assessment model score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='1653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Unlike the previous two webpages, the third one is high-quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' It is carefully laid out and informative, and quality score is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='9788, which will help the ranking system raise its ranking position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2 Online Position Changes The case shown in Figure 4(b) comes from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Comparing the two webpages, we can observe that the left webpage (quality score is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='5623) contains a rich amount of information but the right one (quality score is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='2415) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' We not only enhance GAT with the read mechanism but also care- fully design the features for improving the quality assessment on the webpages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In addition, taking into account the particularity of real-world data, we utilize the category of webpage for optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Both input data construction and model calculation are offline, which guarantees the efficiency of the ranking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Extensive offline and online experiments have shown that the ranking system can significantly improve the effectiveness and general usability of the search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In future work, we will explore the heterogeneous GNN architec- ture to model the multiple graph-based information of webpages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' It is interesting to improve the construction method of layout and enhance the representation of nodes/edges with self-supervised contrastive pre-training techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' REFERENCES [1] Armen Avetisyan, Tatiana Khanova, Christopher Bongsoo Choy, Denver Dash, Angela Dai, and Matthias Nießner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' SceneCAD: Predicting Object Align- ments and Layouts in RGB-D Scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 安全国品电子电器政装用品 外饰用品内饰用品养护用品自空游用品正报价 M Opma适8轮胎防摄膜章道询 昆聘特区：车博土 巴洛克 喜普风 香玉儿 迪佳 键科 暖忆 H H H同酮片购买 Q@: 你可自能摄线： 吃米非司酮片后流血 吃完米非司酮片的反应 吃了米非司酮片流血 吃米非司酮片有什么症状 服米非司酮片 吃米非司酮片第二天出血 吃米非司酮片的注意事项 吃米非司酮片会流血吗 开门红 咨询药师 药师微信什公值得买 Q搜索分类/品牌/商品 打开 全部奶价 社区 商品百料 抗事等 要爽玩《魔兽世界：争霸艾泽拉斯》 CPU 鲁118-88-50 +美注 RYZEN WORLDL 于一 量 家宝业，额心后上靠玩安成的原国，可情不远 大本身视普世养料 MAGB550MMORTARWiFI迫击炮+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' ￥2392 1899 抢 市场价 值得买APP专享价代公值得买 Q搜索分类/品牌/商品 打开 泡好价 全部好价 社区 商品百料 捷惠等 甲全部评论(119) 13心 3068-08-0 心 ALARE 心 3008488-30 爱有年天 4心 MAGB550MMORTARWiFI迫击炮+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' X ¥2399 1899 抢 市场价 值得买APP专享价S搜狐网 晴/1°三 如何搭迪和自己不熟的女同事-如何搭让 如何搭训技巧之一：微笑地说出对方的名字 对于安生而言，如果一入男士非常绅士地对 她微笑，并具当着她的面，自然友好地叫出 了她的名字，她肯定会感到惊访，但随之而 来的更多是欢喜。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='这种搭训会让女生瞬时记 住自己，并且留下较好的印象。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 因为你微笑对她，她也会回以礼貌的微笑 然后她会反问：“你怎么知道我的名学”。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='就 解中度过愉悦的时光。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='彼此都给对方以缸服 的感觉，这为下次的聊天或相聚做好 垫。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 讨论5女相100男，徐州一相亲大会男女比例失调.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='.> 男 女 我来说两句 C知乎 Q中国灵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content='. 下载App 注册登录 不认识的同事（女）如何搭训？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 关注问题 写回答 1个回答 小白兔 谢邀，建议先从她认识的人入手，比如拿联系方 式，然后可以先切入聊天 发布于2020-11-1421:17 一费同 评论 智能消费 新浪潮 创卷调研 广告 相关推荐 男朋友打游戏正确的处理方式，这个 女朋友不能要了 微博的广告 如何去搭陌生人 haoyunlai2188的文章 ApP内打开Layout-aware Webpage Quality Assessment SIGKDD ’23, August 06–10, 2023, Long Beach, CA [3] Yoshua Bengio, Aaron Courville, and Pascal Vincent.' metadata={'source': 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Graph-based Deep Generative Modelling for Document Layout Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In ICDAR Workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' [5] A Caro, Coral Calero, Ismael Caballero, and Mario Piattini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' Data qual- ity in web applications: A state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FLT4oBgHgl3EQfti_6/content/2301.12152v1.pdf'} +page_content=' In IADIS International Conference WWW/Internet, Vol.' metadata={'source': 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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 +References +Ahmed Abdelali, Francisco Guzman, Hassan Sajjad, +and Stephan Vogel. 2014. +The AMARA corpus: +Building parallel language resources for the educa- +tional domain. +In Proceedings of the Ninth Inter- +national Conference on Language Resources and +Evaluation (LREC’14), Reykjavik, Iceland. Euro- +pean Language Resources Association (ELRA). +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. +2020. Language models are few-shot learners. In +Advances in Neural Information Processing Systems, +volume 33, pages 1877–1901. 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The Goldilocks principle: Reading +children’s books with explicit memory representa- +tions. +Jordan Hoffmann, Sebastian Borgeaud, Arthur Men- +sch, Elena Buchatskaya, Trevor Cai, Eliza Ruther- +ford, Diego de Las Casas, Lisa Anne Hendricks, +Johannes Welbl, Aidan Clark, et al. 2022. +Train- +ing compute-optimal large language models. arXiv +preprint arXiv:2203.15556. +Frank Keller. 2010. Cognitively plausible models of +human language processing. In Proceedings of the +ACL 2010 Conference Short Papers, pages 60–67, +Uppsala, Sweden. Association for Computational +Linguistics. +Pierre Lison and Jörg Tiedemann. 2016. +OpenSub- +titles2016: Extracting large parallel corpora from +movie and TV subtitles. In Proceedings of the Tenth +International Conference on Language Resources +and Evaluation (LREC’16), pages 923–929, Por- +torož, Slovenia. European Language Resources As- +sociation (ELRA). +Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, +Mandar Joshi, Danqi Chen, Omer Levy, Mike +Lewis, Luke Zettlemoyer, and Veselin Stoyanov. +2019. +RoBERTa: A robustly optimized bert pre- +training approach. Computing Research Repository, +arXiv:1907.11692. +Brian MacWhinney. 2000. The CHILDES project: The +database, volume 2. Psychology Press. +Colin Raffel, Noam Shazeer, Adam Roberts, Kather- +ine Lee, Sharan Narang, Michael Matena, Yanqi +Zhou, Wei Li, and Peter J. Liu. 2020. +Exploring +the limits of transfer learning with a unified text-to- +text transformer. Journal of Machine Learning Re- +search, 21(140):1–67. +Julian Salazar, Davis Liang, Toan Q. Nguyen, and Ka- +trin Kirchhoff. 2020. Masked language model scor- +ing. +In Proceedings of the 58th Annual Meeting +of the Association for Computational Linguistics, +pages 2699–2712, Online. 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CRC Press. +Thomas Wolf, Lysandre Debut, Victor Sanh, Julien +Chaumond, Clement Delangue, Anthony Moi, Pier- +ric Cistac, Tim Rault, Rémi Louf, Morgan Funtow- +icz, Joe Davison, Sam Shleifer, Patrick von Platen, +Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, +Teven Le Scao, Sylvain Gugger, Mariama Drame, +Quentin Lhoest, and Alexander M. Rush. 2020. +Transformers: State-of-the-art natural language pro- +cessing. In Proceedings of the 2020 Conference on +Empirical Methods in Natural Language Processing: +System Demonstrations, pages 38–45, Online. Asso- +ciation for Computational Linguistics. +Susan Zhang, Stephen Roller, Naman Goyal, Mikel +Artetxe, Moya Chen, Shuohui Chen, Christopher De- +wan, Mona Diab, Xian Li, Xi Victoria Lin, Todor Mi- +haylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel +Simig, Punit Singh Koura, Anjali Sridhar, Tianlu +Wang, and Luke Zettlemoyer. 2022. +OPT: Open +pre-trained transformer language models. Comput- +ing Research Repository, arXiv:2205.01068. + diff --git a/CNFKT4oBgHgl3EQfXy51/content/tmp_files/load_file.txt b/CNFKT4oBgHgl3EQfXy51/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0ed0c0c63aa011855a8c81d13c59dff321b02ea --- /dev/null +++ b/CNFKT4oBgHgl3EQfXy51/content/tmp_files/load_file.txt @@ -0,0 +1,304 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf,len=303 +page_content='Call for Papers - The BabyLM Challenge: Sample-efficient pretraining on a developmentally plausible corpus https://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/ 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'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'} +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 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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': 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'/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'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'} +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': 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0000000000000000000000000000000000000000..61f564cb13e4a359604c205918f9858ab5a65a2f --- /dev/null +++ b/G9E0T4oBgHgl3EQfhgFS/content/tmp_files/2301.02432v1.pdf.txt @@ -0,0 +1,1388 @@ +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. 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DOI:10.1145/ +3458817.3476176. +Prepared using sagej.cls + diff --git a/G9E0T4oBgHgl3EQfhgFS/content/tmp_files/load_file.txt b/G9E0T4oBgHgl3EQfhgFS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9d25799e09faa9ded18b78e0d5fe2f6388cc387 --- /dev/null +++ b/G9E0T4oBgHgl3EQfhgFS/content/tmp_files/load_file.txt @@ -0,0 +1,891 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf,len=890 +page_content='Myths and Legends in High-Performance Computing arXiv preprints ©The Author(s) 2023 Reprints and permission: sagepub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='uk/journalsPermissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='nav DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1177/ToBeAssigned www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='sagepub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' While some laws end, new directions open up, such as algorithmic scaling or novel architecture research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, these myths are rarely based on scientific facts but often on some evidence or argumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In fact, we believe that this is the very reason for the existence of many myths and why they cannot be answered clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We would like to see our collection of myths as a discussion of possible new directions for research and industry investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Keywords Quantum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' zettascale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' HPC myths This manuscript is intended for the “CCDSC Special Issue”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Introduction Any human society has their myths and legends—this also applies to the high-performance computing (HPC) community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' HPC drives the largest and most powerful computers and latest computing and acceleration technologies forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' One may think that it’s scientific reasoning all the way down in such an advanced field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Yet, we find many persistent myths revolving around trends of the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Since it’s late 2022, we started our analysis by asking the all-knowing intelligence ChatGPT “Create myths or legends in high performance computing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' So, even the smartest of internet parrots (Bender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021) that was itself created with massive high-performace computation running on a large accelerator system still has a long way to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Thus, we fall back to reasoning among the authors of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 1: Quantum Computing Will Take Over HPC!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Numerous articles are hyping the quantum computing revolution affecting nearly all aspects of life ranging from quantum artificial intelligence to even quantum gaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The whole IT industry is following the quantum trend and conceives quickly growing expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The actual development of quantum technologies, algorithms, and use- cases is on a very different time-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Most practitioners would not expect quantum computers to outperform classical computers within the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Yet, we have constantly been surprised by advances in device scaling as well as, more recently, artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Thus, the fear of missing out on getting rich is driving the industry to heavily invest into quantum technologies pushing the technology forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Yet, the nature of quantum computation poses some severe limitations: First, reading unstructured data into a quantum state seems very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Hochleistungsrechnersyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', Universit¨atstrasse 6, 8092 Z¨urich, Switzerland Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls [Version: 2017/01/17 v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='20] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='02432v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='DC] 6 Jan 2023 2 arXiv preprints Terabit/s—many orders of magnitude more (Hoefler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Furthermore, once a quantum state is constructed, it can often be “used” only once because measurements destroy superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' A second limitation stems from the lack of algorithms with high speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Most algorithms achieve quadratic speedups for a wide range of use-cases using amplitude amplification at their core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' While this technique is extremely versatile and can search any unstructured quantum state (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Thus, it seems unlikely that quantum computation is going to replace a significant fraction of traditional HPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It is more likely that it will start as quantum acceleration with a small set of use-cases that may grow in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' To determine which use-cases can realistically benefit from quantum acceleration, resource estimation techniques (Beverland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022) become crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Furthermore, it is crucial to guide the resources we invest into the right directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x When will quantum computing be commercially profitable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y What will be the first useful algorithm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z What will be the next break-through area enabled by a new quantum algorithm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 2: Everything Will Be Deep Learning!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Simultaneously with the quantum hype, we are in the midst of the deep learning revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Indeed, in recent years there has been a plethora of papers replacing traditional simulation methods, or whole computational kernels with data-driven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Most of those employ deep neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Impressive results fire up expectations equally high to the quantum world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Data-driven weather and climate predictions apparently beat the best models (Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022) and output data can be compressed by three orders of magnitude (Huang and Hoefler 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Similar successes are touted in literally any application area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' There is no doubt that deep learning models can learn to approximate complex functions used in scientific simulations in a specific input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In fact, any result can be skewed into any of the extremes (Hoefler 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', Rasp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Brenowitz and Bretherton (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' One wonders sometimes whether the latter could have been simplified in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' A new line of research is addressing this by using neural architecture search for such models (Kasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Overall the jury is still out as to which extent surrogate models can replace first-principles simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, one thing is clear: there is a whole spectrum of simulation tasks (Lavin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021)—ranging from ones where exact mathematical expressions are not available in the first place (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', contribution of specific vegetation to weather dynamics) and learning it from data could not only be more efficient but also more accurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' to those where utmost accuracy and precision guarantees are required and can only be provided by specialized error-controlling numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Will ML models replace or just augment traditional simulations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Where will ML models fail to deliver?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z How can we classify (pieces of) an application as ML- acceleratable or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 3: Extreme Specialization as Seen in Smartphones Will Push Supercomputers Beyond Moore’s Law!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' AI, like Stable Diffusion, is now in the palm of everyone’s hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The main purpose of such a composition is to achieve low power for longer battery life while maintaining acceptable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, such a claim misses the point in the analogy, and entirely ignores multiple drawbacks of such an approach as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In fact, the only successful “accelerator” in the recent history of HPC is a GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The reason for the acceleration is primarily that the majority of the HPC workloads are memory bandwidth bound (Domke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Thus, modern Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' So, to claim that multitudes of special accelerators will constitute a supercomputers is stretching the success of GPUs somewhat unfoundedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In fact, there are mainly three reasons why the plethora of customized accelerated hardware approach would fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Now, weak scaling to large number of nodes require that the workload can be subdivided to achieve extremely good load balancing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', (amount of work) / (processing capability) is uniform among all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Practically all successful ‘accelerated’ supercomputers and their applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', GPU machines such as Frontier, follow this pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' For smartphones, the major cost of the phone is not the SoC but rather in the peripherals such as screen, camera, flash memory, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', and the battery life is premium in the cost metric so extra cost incurred by dark silicon may be tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' For supercomputers, however, the major cost of the machine is the processors themselves, dominating over 50% of the overall CapEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' So unless the acceleration could benefit some major proportion of the workload, dark silicon would become an intolerable waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The third reason is software and productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Unless the accelerator usage is extremely easy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', hidden under a set of very simple APIs, expecting the programmers to adopt an arcane programming model is not viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Will extreme heterogeneity happen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Are supercom- puter workloads worth extreme specialization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z When will we have production supercomputers with more than one accelerator type?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' ∗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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, such improvements usually allow increase in the number of nodes and/or processor units, thus helping to push weak scaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' as such, in terms of algorithmic scalability, weak scaling is still the dominating factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls 4 arXiv preprints Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Classification of Compute Kernels and Supercomputing Architecture Myth 4: Everything Will Run on Some Accelerator!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Indeed, for GPU-based machines that would be an assumption, lest the extra investment will not make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, one could question, would some superchip such as GPUs largely replace the CPUs, the latter be degraded to second class citizens?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It is not trivial as it may seem, as such statements are rather dogmatic and not based on candid analysis of the workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' From the hardware perspective, workloads can be largely divided into three classes, (C) compute bound, (B) memory bandwidth bound, and (L) memory latency bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Over time, supercomputer architectures have evolved in an attempt to cover all three in effective ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Up until the 90s, special-purpose vector machines such as Cray and NEC SX accelerated largely (B), and (C) to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This was largely due to the dominant workload that was CFD which was largely (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', DoE ASCI Red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Individual processors were mediocre in performance but attained performance via massive parallelism, exercising weak-scaling, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The roadblock was power/performance using conventional CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, this is not the only possible combination, and other configurations have not been properly explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='. 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In such a machine, the CPU would cover workloads (B) and (L), while the matrix accelerator will cover (C), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' For further acceleration of (L) workloads, there is a limit to acceleration, such as molecular dynamics that require strong scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The best strategy seen for such workloads is fully customized data pipelines such as Anton (Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2008) with hardware design time synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' One could almost mimic such customization with cost but make it programmable by FPGAs or CGRAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' As such, in such a machine, (B) will be covered by CPUs, while (C) and (L) will be covered by a ‘strong scaling accelerator’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' As we observe here, we find that we have not even covered the possible configurations of divergence/convergence of Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Will CPUs become pure “servants” to the accelerators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Are accelerators actually more than just better balanced processors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z Will reconfigurable accelerators see a renaissance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 5: Reconfigurable Hardware Will Give You 100X Speedup!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In a “fool me once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='..” fashion, one accelerator in particular has taken the HPC community by storm with lofty promises of 100x speedup (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2010) ever since the first ported matrix-multiplication by Larsen and McAllister (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Consequently, 164 systems of the TOP500 list utilize accelerators from NVIDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' One positive aspect is that that much code has been “modernized”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', rewritten in CUDA or languages and frameworks promising portability to utilize new devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' But the open question is how portable are these modernized codes really?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Can they run seamlessly on all new devices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The global FPGA market was recently valued at about one-third of the global GPU market (Allied Market Research 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, so far this has not materialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Unfortunately, we see hurdles in all these areas, which the community and industry might be able to solve with enough time and money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The question whether reconfigurable logic can replace or ament GPUs as accelerators is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' FPGAs will certainly have a harder time due to their high flexibility that comes at a cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Units built from reconfigurable logic are 10–20x less energy and performance efficient in silicon area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This issue can be addressed by hardening certain blocks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', floating point units, as some FPGA companies do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' But if now 90% of the chip are hardened units, are those devices just GPUs with a less mature ecosystem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Will the HPC community embrace FPGAs as alternatives to GPUs in large-scale production systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Can the community afford a “Fool me twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='..” moment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z Will CGRA-style reconfigurable dataflow accelerators take the place of FPGAs to compete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 6: We Will Soon Run at Zettascale!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Maybe FPGAs are the way to zettascale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' With Aurora still under construction, Intel ignited the debate about zettascale in late 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This proposition needs to be addressed, and we try to put their claims into perspective and predict a realistic timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' But first we have to distinguish the terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Before we extrapolate, we look at historical trends by Strohmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The HPC community achieved 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='068 teraflop/s with Sandia/IBM’s ASCI Red in summer 1997, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='026 petaflop/s with Los Alamos/IBM’s Roadrunner in summer 2008, and achieved (unofficially) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='05 exaflop/s in spring of 2021 with China’s OceanLight system and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1 exaflop/s with OakRidge/HPE’s Frontier in summer 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Not only do 11 and 13 years lie in between these achievements, respectively, but also multiple megawatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' ASCI Red consumed “only” 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='850 MW, Roadrunner increased that to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='35 MW, and OceanLight and Frontier now consume 35 MW and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1 MW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This and Figure 2 show that the energy efficiency of modern chips cannot keep up with the demand for increasing compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Hence, this 1,000x in performance comes at the cost of 3–5x in power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' and reformulated: the energy efficiency to perform fp64 operations needs to increase by 200–350x, from ≈50 to over 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='000 Gflop/s Watt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls 6 arXiv preprints Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Historical fp64 power efficiency [in Gflop/s Watt ] extrapolated until 2038 to put Intel’s zettaflop/s claims into perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 125x improvement in 10 years, and all of that while other big players slowly acknowledge the end of practical silicon scaling laws (White 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' If we believe the IEEE IRDS™ (2021) roadmap, we might gain 5x in power density (optimistically rounded from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='27x) by 2034 at 7 ˚A compared to 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This leaves 25x, which we could split into 5x from increased transistor count per chip and 5x from increased node count per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Can we cool the former, yes (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021), and can we interconnect the latter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Sure, but doing so, at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', currently fp16 tensor cores demonstrate an 8x advantage over fp64 vector units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Lowering the precision further from fp16 to 3-bit operands could allow for another 5x improvement (Frantar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022), but only if the industry (and HPC community) sees the need for adding these low-precision units, as we discuss in Myth 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Can Intel or anybody else pull it off before then?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Only time will tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Will we reach zettaflop/s performance or will fp64 lose relevance before?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Will we continue to build more power-hungry supercomputers as we did in the past?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z Which one will happen first: zettascale, practical quantum advantage, or all internal combustion-based engines cease to be produced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 7: Next-Generation Systems Need More Memory per Core!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This was mainly due to the quick growth in the number of cores while the performance per core was stagnating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Simplistic arguments along the lines of “we need more of X” seem to have a solid tradition in our community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' For example, the HPC community spent the first decades to hunt more floating point computations per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Recently, a demand for larger and faster memory replaced this main goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The community nearly made a complete 360-degree turn, with Haus (2021) saying “computation is free” and Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (2021) showing “data movement is all you need”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Some even argue that this turn was taken too late due to the fixation on flop/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', ratio between the different resources such as memory size/bandwidth and compute (Czechowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' These ratios usually shift with chip technology and architectural choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This led to the recent data movement crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Newly emerging optical off- chip connectivity, see Myth 8, as well as 3D integrated memory (Domke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022) shifts the balance again and may alleviate many of these aspects, at least at the scale of a single chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It seems key to understand the malleability of application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', which resources can be traded for which other resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', memory capacity for computation bandwidth using recomputation or caching as techniques).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls Matsuoka, Domke, Wahib, Drozd, Hoefler 7 Here, specifically I/O complexity analysis is a tool to deeply understand this trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Once all trade-offs are understood, requirements models (Calotoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2018) could be used to fix trade-offs into designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' These models could then inform architectural choices as well as hardware developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', battery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' While those designs were initially limited to very narrow application domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', radio signal, audio, or video processing), embedded devices have recently been expanded towards more diverse workloads (“apps”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x When will the current “data movement” focus end?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y What will be the next bottleneck resource?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z Will our community be able to adopt a performance modeling discipline to discuss bottlenecks scientifically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 8: Everything Will Be Disaggregated!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' To stop the waste of memory resources, the academic com- munity is advancing on the Silicon Photonics front (Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022) and industry is pursuing scale-out technologies (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022), such as Compute Express LinkTM (CXL), a cache-coherent interconnect for data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' But a few players seem to push the idea over the edge with their plans to disaggregate everything (NTT R&D 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' As Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Generally, we see two remaining challenges for a broad adoption of Silicon Photonics and all-optical interconnects: low-cost manufacturing and optical switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The latter challenge is less obvious for the uninitiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Current electrically switched networks can operate in “packet switching” mode to effectively lower the observable latency and utilize the available link bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The alternative to this mode is “circuit-switching” and it was abandoned by the HPC community long ago in favor of packet-switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' For HPC, an extensive or extreme disaggregation yields another challenge, specifically the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Photons travel at a maximum speed of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='3 ns/m in hollow fibers (or slower in other transport media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' At 3–4 m distance, the photon travel time alone exceeds the first-word access latency of modern DDR memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Therefore, if main memory would be disaggregated beyond rack boundaries, it will become noticeable for memory- latency sensitive applications, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The more sensible solution, in line with Myth 7, for future HPC systems are smaller node-local memory configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Will CXL be deployed widely in HPC?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Will large- scale supercomputers be disaggregated beyond rack- scale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z Should we disaggregate main memory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 9: Applications Continue to Improve, Even on Stagnating Hardware!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Modernizing hardware, with Silicon Photonics, Tensor Cores, or simply shrinking transistors, has too long been the primary method of accelerating legacy software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The remaining hardware improvements came from architectural innovations, such as deeper cache hierarchies, the migration to more specialized architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Unfortunately, we are no longer seeing the consistent technology scaling that Moore observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Similar reports attribute a significant portion of the performance Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='arXiv preprints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='higher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='order AMR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='10 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Effective Sustained Speedup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content="Algorithmic Moore's Law Examples " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Sustained Speed in Gflop/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Combustion Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='(Complex Kinetics) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Combustion Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='(CFD) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='COSMO Climate Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Fusion Energy Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='(Global MHD) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Moore’s Law ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='Fusion Energy Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='(Micro-turbulence) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='improved ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='linear solver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='ARK integrator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='complex chem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='AMR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='semi-implicit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='high-order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='elements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='gyro- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='kinetics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='delta-f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' magnetic coordinates improved electron models low Mach auto-code high order improved explicit/implicit solvers Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Examples of “Algorithmic Moore’s Law” for different areas in HPC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' That being said, those limits might not be as clear and quantifiable as the limits on hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x As the performance improvements from hardware technologies drop, should the HPC community dramat- ically increase the investment in software?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Will the “Algorithmic Moore’s Law” end soon as well?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z To what extent is the HPC community willing to refactor/rewrite legacy codebases when/if hardware stagnates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 10: Fortran Is Dead, Long Live the DSL!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Applications might have limits, but what about languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' How often have we heard “Fortran is dead, long live X”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Slogans like this have been resonating in the community for nearly 40 years (Post 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' X has been everything from C to C++, and more recently Python or Domain-Specific Languages (DSLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Yet, Fortran remains in wide use in important communities such as weather and climate even for newly written codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Other languages, such as COBOL were indeed replaced with more modern alternatives such as Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Why is this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Are some parts of our community just stubborn to follow the youngsters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Or are old languages not necessarily bad for the task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Indeed, Fortran is a very well designed language for its purpose of expressing mathematical programs at highest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It seems hard to replace it with C or other languages and outperform it or even achieve the same baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' This may be due to the highly optimized Fortran compilers or the limited language features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', no pointer aliasing) that enable more powerful optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Fortran and other general-purpose languages remain competitive with many DSLs on CPUs (Ben-Nun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2022) and are recently also adopted to GPUs, albeit often less elegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' General-purpose portability approaches such as SYCL (Keryell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2015), also powering Intel’s oneAPI, or OpenMP provide flexibility as well as some portability across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' High-productivity general-purpose languages are hard to accelerate in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' For example, Python’s flexibility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', monkey patching and flexible typing) disables many static optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, when restricting the syntax to high-performance Python (much of NumPy), then optimizations become simpler (Ziogas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Any language becomes more complex over time—Fortran 66 evolved into the complex Fortran 2018 language standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Similar trends affect DSLs that are widening their scope over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Do we require this generality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' If yes, then DSLs are doomed to fail or they morph into general-purpose languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Then the question remains—where should the portability layer be located?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2019)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x When will programmers stop using Fortran for new applications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Will we ever have more application codes written in DSLs than general-purpose languages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z What will be the next big DSL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 11: HPC Will Pivot to Low or Mixed Precision!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Lowering this precision can save costs Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls Matsuoka, Domke, Wahib, Drozd, Hoefler 9 but may reduce accuracy of the results and, in the worst case, break the application (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Some processors offer mixed precision tricks at the hardware level in the form of instructions with low precision inputs but higher precision accumulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 2021), or even if to use an entirely different (not IEEE-754) way to represent numbers (Gustafson and Yonemoto 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Exploding and vanishing gradients, outlier weights, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', made training large deep neural nets require extra effort to stabilize (incurring corresponding overhead) or just did not converge at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The next generation of devices came with bfloat16 format—same 16 bits, but more bits allocated to range, less for precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It worked better, but still once in a while a model would collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Finally, the recent generation of GPUs came with a 19-bit numeric format, misleadingly called TensorFloat-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Now that mixed precision is a de-facto standard in the AI domain, more hardware support is being implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' So far there is no general clarity on the limits—how few bits can we get away with in different HPC areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The following factors in particular are important to consider as we move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' A fully transparent solution for the problem is to simulate higher precision using low precision operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', as shown by Ootomo and Yokota (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It is not clear however if this incurred overhead is an acceptable price that HPC agrees to pay for remaining in higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=', as demonstrated by Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' One main limitation of that method is the reliance on automatic differentiation (AD) to track error propagation, which is not practical for large codebases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x Is the HPC community ready (or already late?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=') to react to the new low precision formats driven by deep learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y Will HPC navigate itself into a high-precision niche?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z When, if ever, will the industry drop fp64 support?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Myth 12: All HPC Will Be Subsumed by the Clouds!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The rapidly advancing AI and new precision options has reignited the cloud discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The question whether clouds will subsume supercomputing has been ongoing for more than a decade, since the late 2000s Deelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (2008), but remains inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' It is not surprising that cloud providers offer exactly the same performance as on-prem supercomputing centers in practice De Sensi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' After all, they simply buy the same hardware!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Thus, this discussion is more of a fiscal argument with an interesting economy-of-scale twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' There are actually bi-directional aspects to the cloud-vs- supercomputer argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We must look at both aspects, and it is in fact the latter where such subsumption may happen or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Indeed, many major supercomputers are already facilitating cloud features, so that they are effectively clouds themselves, and interoperable with commercial clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' However, this assumes that there is already a supercomputing resource facilitated for themselves, and does not directly affect the subsumption argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Certainly, there are now multiple cloud services that facilitate virtual compute clusters in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In fact, for enterprise IT infrastructures, there has always been a swing between on-prem and public clouds, largely Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls 10 arXiv preprints driven by economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' The R&D expenditures of hyperscalers in IT now outclass the government investments, and increasingly the hyperscalers are investing in high end computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' If the commercial cloud hyperscalers can work out the scale of economy in their own hardware manufacturing to the extent that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' it could build and operate large scale HPC infrastructures cheaper than on-prem supercomputers of any size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' then the swing could totally happen towards full subsumption— although somewhat unlikely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' this could compromise the ability to cover some of the traditional HPC workloads that do not meet main industrial needs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' such as the requirement for dense 64 bit linear algebra capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' We close with these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' x What could be a defining development to decide between cloud and on-prem HPC?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' y When will more than half of the HPC cycles be spent in the cloud?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' z Will on-prem systems be a niche or remain with a significant fraction of HPC cycles spent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' While we present them in an exaggerated and humorous way, many of those myths form the core of thinking in our community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Some may be more divisive than others but it seems that many are hard to answer definitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Maybe some points will settle in the future while others will not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Yet, their sheer importance mandates a serious treatment in order to help guide future directions for academic research but also industry and government investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' References Allied Market Research (2020) Graphic Processing Unit (GPU) Market by Type (Dedicated, Integrated, and Hybrid), Device (Computer, Tablet, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' https://tesla-cdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='thron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='com/static/ MXMU3S_tesla-dojo-technology_1WDVZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' White MJ (2022) Nvidia says falling GPU prices are ’a story of the past’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' https://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Ziogas AN, Schneider T, Ben-Nun T, Calotoiu A, De Matteis T, de Fine Licht J, Lavarini L and Hoefler T (2021) Productivity, Portability, Performance: Data-Centric Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' ISBN 978-1-4503-8442-1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='1145/ 3458817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='3476176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content=' Prepared using sagej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} +page_content='cls' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfhgFS/content/2301.02432v1.pdf'} diff --git a/GNE1T4oBgHgl3EQfXAT2/content/tmp_files/2301.03123v1.pdf.txt b/GNE1T4oBgHgl3EQfXAT2/content/tmp_files/2301.03123v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a644e30ef4e48f754cb56706c51ec095861ef02 --- /dev/null +++ b/GNE1T4oBgHgl3EQfXAT2/content/tmp_files/2301.03123v1.pdf.txt @@ -0,0 +1,1030 @@ +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 + diff --git a/GNE1T4oBgHgl3EQfXAT2/content/tmp_files/load_file.txt b/GNE1T4oBgHgl3EQfXAT2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a17359e36dbd7c96a8e1da59b21ab48bde30a82 --- /dev/null +++ b/GNE1T4oBgHgl3EQfXAT2/content/tmp_files/load_file.txt @@ -0,0 +1,620 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf,len=619 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='03123v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The techniques are general enough to consider their applications in many other frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 1 Introduction In the geometric world, categorical colimits can be thought of as a general way of expressing gluing of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For example, any scheme is the colimit, in the category of locally ringed spaces, of the components of any of its affine coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' to the category of commutative rings with unit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 1 X → CRing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' and in many cases they coincide with their scheme-theoretic analogues on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A brief summary of this non-standard approach will be provided in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The situation is not much better if we attempt to compute colimits of schematic spaces as described in the previous paragraph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In this paper, we will study this cylinder space in its most general formulation, replacing CRing for any category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The category of finite posets with a C-values structure sheaf will be constructed and called the category of C-data, denoted C -data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For this purpose, we will need to consider D -data with D being a strict 2-category, which also proves useful in other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Our language of choice will keep things analogue to the category of ringed posets that we use as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The result is a morphism of ringed spaces π: S → (X, π∗OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 3 Schematic spaces were first introduced in [6] and studied in [5], [8] or [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' but schematic spaces do not model, for example, algebraic spaces, for which the associated locally ringed space does not preserve enough useful algebraic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' There are also "geometric" arguments for considering ringed posets as the basis for our combinatorial models over other, more classical, alternatives like simplicial schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' While perhaps not the most enlightening approach, it will be convenient for our purposes to consider the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Assume for simplicity that all stalk rings of our ringed posets are Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A finite ringed poset X is a (finite) schematic space if For any x ≤ y, the morphism rxy : OX,x → OX,y is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For any t ≤ x, y, the morphism OX,x ⊗OX,t OX,y → � z≥x,y OX,z is faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A simple descent argument for faithfully flat morphisms shows that OX,x ⊗OX,y OX,y ≃ OX(Ux ∩Uy) for all t ≤ x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If x = y, this condition implies that the restriction morphisms rxy of any schematic space are flat epimorphisms of rings, hence local isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 4 Let SchFin denote the category of schematic spaces and morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' All ringed posets and morphisms will be considered schematic unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Heuristically, the restriction maps of X being flat epimorphisms implies that the information in X can be recovered from Spec(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The other schematicity conditions can be shown to be equivalent to the existence of a certain map πX : Spec(X) → X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' to X being essentially a "finite model" of Spec(X) in a topological sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A morphism f : X → Y is said to be a qc-isomorphism if Spec(f) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' To study properties P of schematic spaces we will ask for two requisites: A "rigorous" requisite: that P factors through the localization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' that any representative of the qc-isomorphism class of a space or arrow determines is the whole class verifies the property or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In other words, P is geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A "moral" requisite: P can be studied in terms of finite models, without applying the Spec functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In other words, P is discretizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' As an example of this, see the discussion about connectedness in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We denote its fundamental group by πet 1 (X, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In [4] we also showed that qc-isomorphic spaces have equivalent Galois categories of finite étale covers: the construction is geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' As always, this is just a particular case of more general topos-theoretic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1 The topology of flat immersions We begin by introducing the natural (pre)topology on SchFin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let f : X → Y be a schematic morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We say that f is flat if f ♯ x: OY,f(x) → OX,x is flat for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Such flat morphism is a flat immersion if its diagonal ∆f : X → X ×Y X is a qc-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A flat morphism f is faithfully flat if Spec(f) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' These three types of maps can be characterized in terms of the adjoint pair (f ∗, f∗) for quasi-coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We remark that a flat immersion is, by definition, a flat monomorphism in SchFinqc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' and Spec(f) for such f is a flat monomorphism of locally ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' One can show that qc- isomorphisms are exactly faithfully flat immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In other words, schematic spaces are colimits of (certain) affine schematic spaces via flat immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This class of morphisms was first shown to be important in the context of descent problems in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If f : X → Y is a flat immersions, f ♯ x : OY,f(x) → OX,x are flat epimorphisms of rings for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Recall that an open immersion of schemes is a flat monomorphism (locally) of finite presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' As such, flat immersions are like "open immersions", but without the finite presentation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The reader might notice the analogy with the étale and pro-étale topologies for schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This justifies the following notation: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let X be a schematic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Similarly, we define SchFinwZar to be the "big" site of flat immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' These sites present a number of interesting pathologies that we will describe in more detail in future papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We shall remark a few of them: The category XwZar is not small, only its localization (XwZar)qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5]—;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' but the good news is that it does exist (XwZar)qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' we can sheafify presheaves that factor through qc-isomorphism, which will be enough in all natural situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Endowing XwZar with the natural sheaf of rings, it is possible to show that Qcoh(XwZar) ≃ Qcoh(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This will be done in Sections 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The result will be called cylinder space, denoted Cyl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For instance, let us consider the case of the étale fundamental groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It clearly defines a functor Πét 1 : SchFin → GpdStone 8 valued in the strict 2-category of Stone groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This will be the same as saying that (Πét 1 )op is a 2-sheaf —thus it maps qc- isomorphisms to equivalences—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Such statement amounts to showing that Πét 1 maps a very specific kind of cylinders to 2-colimits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' and in some particular cases, this will even be formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For a given poset X and x ∈ X, let Ux = {x′ ≥ x} denote the minimal open neighborhood of the point X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The following is well-known: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Each sheaf gives a functor defined by its stalks—sections at the minimal open neighborhoods—and restrictions morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For any C, the cateory of C -data is the fibered category over C defined by the Grothendieck construction applied to the previous functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Explicitly: 9 Ob(C -data) = {F not ≡ (X, F) : X ∈ pos and F ∈ [X, C]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' HomC -data((X, F), (Y, G)) = {f : X → Y and f ♯: f −1G → F};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' | − |: C -data → pos is the "underlying poset" structure functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Furthermore, for any F and x ≤ y ∈ |F|, we will denote its "restriction morphisms" by Fxy : F(x) → F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Due to the choice of ⋆ as the final object in pos, we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If C has finite limits (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' colimits), the functor iC has a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' right) adjoint Γ ≡ ΓC : C -data → Cop (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' L) called the sections (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' cosections) functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Explicitly, Γ(F) = lim F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Γ(F) = colim F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The terminology of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5 comes from the equivalence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' One may also interpret sections via projections to the terminal poset π: X → ⋆ by constructing π∗ right adjoint to π−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='7 (Locally representable functors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' As a simple application of this terminology, we will give a "structured" interpretation of the concept of locally representable functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Indeed, let Y : C → [Cop, Set] be the Yoneda embedding for C and Y∗ : C -data → [Cop, Set] -data the—fully faithful— induced functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' One may think of an object in the image of Y∗ as a "locally representable functor".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Note that, if C has finite limits, the sections of such an object are representable by the sections of the original C-datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Additionally, we shall consider the Yoneda embedding for C -data, that is Y ′ : C -data → [C -dataop, Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We simply ask this arrow to induce a commutative triangle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' we define our 2-morphisms to be: HomHomC -data(F,G)(f, g) = � ⋆ if f ≤ g and g♯ = rfg ◦ f ♯ ∅ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We note that this structure generalizes the partial order defined in [7] to study naif homotopy types of ringed posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We also remark that C -data is actually a pos-enriched category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It is easy to check that, if C has finite limits (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' colimits), then C -data has finite colimits (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' limits), described in an analogous way as in the category of ringed posets (or spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We will call this construction the "cylinder functor".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 11 4 The 2-categorical case Let D be a strict 2-category and endow posets with the trivial 2-categorical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Among other possibilities, we shall consider the categories of pseudofunctors X → D and pseudonatural transformations, [X, D];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' pseudofunctors X → D and lax natural transformations, [X, D]Lax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The Grothendieck construction for each of these possibilities now yields, as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2, two different 1-categories, denoted for emphasis as D -data and D -dataLax respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Again, if D is pos-enriched, giving this lax natural transformation amounts to asking that, for |f| ≤ |g|, we only have rfg ◦ f ♯ ≤ g♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Finally, as in the 1-categorical case, and almost by definition, we have: 12 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The left 2-adjoint of iD (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' iLax D ) is, if it exists, the pseudolimit (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' lax limit) of the structure pseudofunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We call it sections (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' lax sections) functor and denote it by Γ ≡ ΓD (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' LaxΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We begin with the explicit description: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It is easy to check that this construction is functorial, thus we have Cyl: (C -data)op -data → C -data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If C = ⋆, hence C -data = pos, the functor Cyl coincides up to natural isomorphism with the lax sections functor of the inclusion i⋆ -dataop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In other words, pos has pos-indexed lax colimits, described by Cyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The converse follows from the same argument: given g: X → Y , the g♯ p are exactly the morphisms fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This is precisely the notion of 2-morphism in posop -dataLax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For any category C, the functor Cyl coincides up to natural isomorphism with the lax sections functor of the inclusion iC -dataop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' C -data has pos-indexed lax colimits and they are described by Cyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The topological part of the proof has been taken care of in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2, so we only need to check that such isomorphism extends to the level of C-valued functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1) where Cyl(X)xpyq = (Xpq)♯ yq, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The converse follows from the same relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Once again, |fp| ≤ |gp| by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2, so we complete the proof by remarking that, for each xp ∈ |X(p)|, the fact that the equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2 holds is equivalent to fp ≤ gp in C -data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This means that, provided that colimits of C -data also exist, there is a natural transformation to the 1-categorical sections: Cyl → ΓC -dataop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' such that (iop C )∗(F) has the same underlying poset as F, but we "replace" each F(p) by the constant datum (⋆, F(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It is obvious that Cyl ◦ (iop C )∗ is also the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The functor Cyl commutes with finite fibered products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Exercise to the reader: it follows from the explicit construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In particular, the cylinder functor restricts to Cyl: SchFinop -data → CRing -data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The next few pages are devoted to characterizing SchFinop-data whose cylinder spaces are schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Given X ∈ SchFinop -data, the natural morphism of ringed spaces Cyl(X) → Γ(X) induces an isomorphism Spec(Cyl(X)) ∼ → Spec(Γ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This follows from the fact that colimits commute with colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A ringed poset X is said to be pseudo-schematic if it has flat epimorphisms of rings as restriction maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A ringed poset X is Mod-affine if π: X → (⋆, OX(X)) induces an adjoint equivalence (π∗ ⊣ π∗): Qcoh(X) → Mod(OX(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We say that X is affine if it is schematic and Mod-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Any ringed poset with a minimum X = Ux is Mod-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If X is pseudo-schematic, Qcoh(X) is a Grothendieck abelian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In particular, if X is also Mod-affine, π∗ is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If X is pseudo-schematic and Mod-affine, the natural morphism OX(X) → � x∈X OX,x is faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It suffices to see that � x∈X Spec(OX,x) → Spec(OX(X)) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Geometrically, this means that the fiber of p via Spec(OX,x) → Spec(OX(X)) is non-empty, so we win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Any ringed poset with a minimum X = Ux is qc-isomorphic to (⋆, OX,x) via the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In the schematic category, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='7 restricts to the usual one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Given X ∈ SchFinop -data whose restriction morphisms are flat immersions, Cyl(X) is pseudo-schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It follows from the construction, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='9 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Now, given X ∈ SchFinop -data and p ∈ |X|, denote by Up the datum induced on the open subset Up ⊆ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We have qc-isomorphisms of ringed spaces πp: Cyl(Up) → X(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' which, composing with the previous projections, induce πt pq : Cyl(Up ∩ Uq) → X(p) ×X(t) X(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' With the technology introduced in this paper, we can only prove the "if" part, which will be the one used in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Now, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6 translates exactly into the conditions of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We only prove the "if" part, which follows from the same results as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Note that, applied to a datum X with X(p) = (⋆, Ap) for all p, Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='10 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='11 restrict to the usual Definition of schematicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' See this in view of Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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—.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Note that U comes equipped with a morphism U → X ≡ iC -data(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' First, we check the condition of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='10: for ∆1, ∆2 ∈ |U|, U∆1 ∩ U∆2 = U∆1∪∆2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Schematicity of Cyl(f) follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='11 and a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The morphism Cyl(f) is flat by the local construction and its diagonal is a qc-isomorphism because, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In the following definition—if appropriate—, one shall consider SchFin as a 1-category with the trivial 2-categorical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let C be a 1-category (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' strict 2-category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' A geometric datum is a functor (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' pseudofunctor) Dat: SchFin → C that maps qc- isomorphisms to isomorphisms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' equivalences);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' in other words, one that factors through SchFinqc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In the discussion that follows, let us assume that C is a 1-category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' the argument also works for 2-categories, replacing isomorphisms by equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4 we saw that there are two natural immersions of any category of C-data into its category (C -data)op -data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In this case, there is a natural transformation between functors in [SchFin, SchFinop qc -data]: ηC : (iop C )∗ → iC -dataop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' For general ringed posets, this natural transformation is induced by the morphisms (⋆, OX,x) → X, which are not schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' That is one of the reasons to consider the localized category, where it is induced by the triangles (⋆, OX,x) ← Ux → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Now, given a geometric datum Dat: SchFinqc → C, we define Dat := Dat∗ ◦ (iop C )∗ Dat ≡ Dat ◦ iC -dataop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' where Dat(X) is the Cop-datum with |Dat(X)| = |X| and structure functor Dat(X)(x) = Dat(⋆, OX,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We say that a geometric datum Dat satisfies internal descent if Γ(ηDat): Γ∗ ◦ Dat → Γ∗ ◦ Dat ≡ Dat is an isomorphism in [SchFin, C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The datum Qcoh: SchFin → Catop satisfies internal descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' which coincides with the description of this pseudolimit in Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This is because base changes by flat epimorphisms of rings satisfy said adjunction (left as an algebra exercise to the reader).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6 (External descent for nerves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='14, Cyl(U) → X is a qc-isomorphism, and since Dat is geometric, one has that Dat(Cyl(U)) ≃ Dat(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In the situation of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6 and thanks to Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5, we obtain that Qcoh(X) ≃ 2-lim∆∈|U| Qcoh(U(∆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In particular, being quasi-coherent is local in the topology of flat immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='8 (External descent for topoi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Similarly, it induces equivalences between respective categories of C-valued sheaves (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' stacks) for any 1-category (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 2-category) that has finite poset-indexed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This is simply a reinterpretation of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6 in terms of the language of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1: sheaves in Sh(SchFinwZar) map qc-isomorphisms to isomorphisms, so they are geometric data in the sense fo this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We shall remark that the analogous equivalence between small topoi does not hold—a priori—because cylinders change the base space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This is similar to what happens in the category of schemes for the set-theoretic topology and the Zariski site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 8 Example: Seifert-Van Kampen Theorem A less trivial application comes from the étale fundamental groupoid—and group—, as promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let us consider the pseudofunctor Πét 1 : SchFin → GpdStone 21 to the 2-category of Stone groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' This follows quite easily in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Before that, we highlight that the category of finite étale covers defined without any detail in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5 can be described in terms of quasi-coherent sheaves of algebras, as done in [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' more precisely: RÉt(X) = � "opposite category of quasi-coherent algebras A such that OX,x → Ax is a finite étale ring map".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1) Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The pseudofunctor RÉt: SchFinqc → Cat defined on objects by1 X �→ RÉt(X) satisfies internal descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' We have to show that RÉt(X) ≃ 2-limx∈X RÉt(⋆, OX,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Since it is a subcategory of the category of quasi-coherent algebras, this follows from Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' The pseudofunctor Πét 1 : SchFin → GpdStone satisfies internal descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Following the notations of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='1 we know that the natural transformation ηRÉt : RÉt → RÉt induces an equivalence after taking sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' where the vertical arrows are isomorphisms after taking sections by the Galois Theorem for fundamental groupoids, hence Γ(ηΦ◦Πét 1 ) an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Finally, since Φ well known to commute with pseudocolimits, one has that Γ(ηΦ◦Πét 1 ) ≃ Φ ◦ Γ(ηΠét 1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' and since this map is an equivalence and Φ is (2-)conservative by [10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='11], Γ(ηΠét 1 ) is an equivalence, which proves the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 1On 1-morphisms, we send each f : X → Y to the inverse image functor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 22 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If a schematic space X satisfies that Πét 1 ((⋆, OX,x)) = {⋆}—the trivial 2-category—for all x ∈ X, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2 yields Πét 1 (X) ≃ 2-colimx∈|X|{⋆} ≃ � Π1(|X|), where the hat denotes the profinite completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let X be a schematic space and {Ui → X} be a covering by flat immersions with associated nerve datum U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Then, the natural morphism 2-colim∆∈|U| Πét 1 (U(∆)) → Πét 1 (X) is an equivalence of topological groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' In other words, the functor Πét 1 is a (co)stack in the topology of flat immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' It follows from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='2 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' One can give the statement of the Theorem in greater generality, for any X ∈ SchFinop-datum such that Cyl(X) is schematic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' and in that case the topological fundamental groupoid of |X| plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' the étale fundamental groupoid of schemes is a costack in the topology of flat monomorphisms and finite covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Finally, we can very easily specialize this result to fundamental groups, which a formulation that we deem more natural than that of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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(∆)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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 ∆ ≤ ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let us denote by x = x∅ the geometric point of X given by this collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 23 Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Then there is an isomorphism of topological groups colim∆∈|U| πét 1 (U(∆), x∆) ∼ → πét 1 (X, x) induced by conjugation the ϕ∆∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Since X is connected, the natural inclusion πét 1 (X, x) → Πét 1 (X) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Let GrStone ⊆ GpdStone be the category of profinite groups, which one may think set-theoretically or as Top-enriched categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Define the datum πét 1 (−, x⋆): |U| → Grop Stone ∆ → πét 1 (U(∆), x∆) whose restriction morphisms given by conjugation with the ϕ∆∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' From this fact and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='4, there are equivalences 2-colim∆∈|U| πét 1 (U(∆), x∆) ∼ → 2-colim∆∈|U| Πét 1 (U(∆)) ≃ Πét 1 (X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Since any equivalence between one-object categories is an isomorphism, the proof ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' If each one of these fibers is simply connected and we assume connectedness, the result recovers the classical one of McCord for π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' 24 References [1] Codara, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' PhD thesis: A theory of partitions of partially ordered sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} +page_content='M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE1T4oBgHgl3EQfXAT2/content/2301.03123v1.pdf'} diff --git a/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/2301.02673v1.pdf.txt b/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/2301.02673v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..be1fd4a636e5b56e48dd5a57507cc157f28cd4e0 --- /dev/null +++ b/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/2301.02673v1.pdf.txt @@ -0,0 +1,940 @@ +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). +[1] S. L. Brunton, and J. N. Kutz. 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Brunton, J. L. Proctor, and J. +N. Kutz, Inferring biological networks by sparse iden- +tification of nonlinear dynamics. IEEE Trans. Mol. Biol. +Multi-Scale Commun. 2, 52. (2016). +[32] M. Ipsen, F. Hynne, and P. G. Sørensen, Amplitude equa- +tions for reaction–diffusion systems with a Hopf bifurca- +tion and slow real modes, Physica D 136, 66 (2000). +[33] It is not possible to remove all numerical derivatives in +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. +[35] J. Burke and E. Knobloch. Homoclinic snaking: structure +and stability. Chaos 17, 037102 (2007). +[36] E. J. Doedel, A. R. Champneys, F. Dercole, T. F. Fair- +grieve, Y. A. Kuznetsov, B. Oldeman, R. C. Paffenroth, +B. Sandstede, X. J. Wang, and C. H. Zhang. AUTO- +07P: Continuation and bifurcation software for ordinary +differential equations. (2007). +[37] K. Champion, B. Lusch, J. N. Kutz, and S. L. Brunton. +Data-driven discovery of coordinates and governing equa- +tions. Proc. Natl. Acad. Sci. U.S.A. 116, 22445-22451 +(2019). +[38] B. Chen, K. Huang, S. Raghupathi, I. Chandratreya, Q. +Du, and H. Lipson, Automated discovery of fundamental +variables hidden in experimental data, Nat. Comput. Sci. +2, 433-442 (2022). +[39] J. Bakarji, K. Champion, J. N. Kutz, and S. L. Brun- +ton. Discovering governing equations from partial mea- +surements with deep delay autoencoders. arXiv preprint +arXiv:2201.05136 (2022). +[40] B. G. Chen, N. Upadhyaya, and V. Vitelli. Nonlinear con- +duction via solitons in a topological mechanical insulator. +Proc. Natl. Acad. Sci. U.S.A. 111, 13004-13009 (2014). +[41] Z. G. Nicolaou, D. J. Case, E. B. Wee, M. M. Driscoll, +and A. E. Motter. Heterogeneity-stabilized homogeneous +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. + diff --git a/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/load_file.txt b/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37b370eee0bcb71aabb7f84e1442cf7e347db808 --- /dev/null +++ b/GtE0T4oBgHgl3EQfzgJq/content/tmp_files/load_file.txt @@ -0,0 +1,686 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf,len=685 +page_content='Data-driven discovery and extrapolation of parameterized pattern-forming dynamics Zachary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou,1 Guanyu Huo,1 Yihui Chen,1 Steven L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton,2 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' First-principle derivations of the underlying transitions can be characterized using bifurcation theory on model sys- tems whose governing equations are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In contrast, data-driven methods for more complicated and realistic systems whose governing evolution dynamics are unknown have only recently been de- veloped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We demonstrate the method on systems of varying complexity, ranging from discrete maps to systems of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To mitigate the impact of measurement noise, we also develop a weak formulation of SINDyCP and assess its performance on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In this work, we leverage the SINDy model discovery framework to understand parametric dependencies and underlying bi- furcations in pattern forming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Specifically, we develop the SINDY with control parameters (SINDyCP) to discover such parameterized dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Despite the success of pattern-formation theory in modeling complex dynamics, ongoing challenges remain in applying such model equations more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' First- principle derivations and the computation of normal- form parameters in terms of physical driving parameters are tedious, costly, and error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A variety of extensions of the SINDy approach have been developed since its in- troduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' SINDy can also learn to disambiguate between parametric dependency and governing equations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Our approach is implemented in the open- source PySINDy repository [23, 24], enabling other pow- erful methods to be used in conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In particu- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='02673v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Schematic of the SINDyCP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sparse regression is applied on the library coefficients ξ to identify a parameterized governing equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' lar, we develop and assess a weak formulation [25–28] of SINDyCP, which shows excellent performance on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Building the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Ginzburg-Landau exhibits a stunning variety of pat- terns, depending on the bifurcation parameters b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We generate four trajectories with parameters val- ues (b, c) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='75), which exhibit differing dynamical phases, corresponding to amplitude turbulence, phase turbulence, stable waves, and frozen spiral glasses, respectively [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' again, there are four trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We then de- termine the temporal derivative ˙X for each sample point, either through numerical differentiation or through direct measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In the SINDYc approach, we augment the li- brary dependence with an external control signal U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=', Θ = Θ(X, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The library terms are typically determined by appending the control variables to the dependent vari- ables and again forming polynomials and derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In the case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2) Crucially, all SINDy methods employ sparse regression (with appropriate regularization) to determine a sparse set of nonzero coefficients ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Such sparsity is expected in physically-relevant dynamics and produces parsimonious and interpretable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A significant challenge arises when applying the tradi- tional SINDYc to control parameters in PDEs with ex- isting implementations such as PySINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The matrix of library terms Θ is traditionally formed by computing all polynomial combinations of spatial derivatives of the dependent and control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' However, since the con- trol parameters are spatially constant, the spatial deriva- tives will vanish identically, leading to a singular matrix Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By distinguishing the feature and parameter library dependencies with this SINDyCP ap- proach, we can construct much more targeted and well- conditioned libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (1) in Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Beyond weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—SINDyCP enables discovery of nonlinear corrections to weakly nonlin- ear theory directly from data that can be gathered in pattern-formation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' To illustrate this result, we implement an in silico experiment of the Belousov- Zhabotinksy chemical reaction system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' While the score decreases modestly as µ increases, the model remains very accurate (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Corrections to the weakly nonlinear theory of the Oregonator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (a) R2 score for the parameterized SINDyCP model on test trajectories collected at the param- eter values used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (b) Corrected normal- form parameter values relative to the weakly nonlinear values b0 and c0 as a function of the bifurcation parameter µ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (c) Limit cycles in the homogeneous system exhibiting the highly- nonlinear canard explosion with increasing µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' on all the testing trajectories, accounting for 99% of the variation in the data in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' A nonlinear change of coordinates transforms the dis- covered model into the normal-form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (1) with parameter-dependent values of b(µ) and c(µ) and small quintic corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2(b), the pa- rameters vary with µ, representing additional corrections to the weakly nonlinear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This variation becomes extreme for µ1/2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='35, which we were able to discover via the implicit version of SINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' In fact, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='39, where the weakly nonlinear theory breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDyCP model reflects this breakdown and enables the development of higher-order corrections to account for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—The weak formulation utilizes in- tegration against compactly supported “test functions” to defined the SINDy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The weak method shows excellent performance for noisy data, owing to its ability to minimize the need for computing numerical deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Rather than forming samples (rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t)X(ν) i (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By moving derivatives off of the data and onto the test functions via integration by parts, wν ik = (−1)|ν| � Ωk φ(ν)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t)Xi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' t) dDxdt, (6) the weak method significantly reduces the impact of mea- surement noise on the SINDy library and improves the fit results [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' De- tails about our efficient numerical implementation are available in the Supplemental Material [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Our implementation thus rescales the weak features for the temporal derivatives by the same volumetric factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We then generate two new sample tra- jectories to use as testing data, with b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5 and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Figure 3(a) shows the re- sults for the R2 score on the test trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDyCP fit also requires a sufficient amount of data to identify governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Unlike the trajectories in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1, the parameters for trajectories were randomly (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Performance of SINDyCP for the fit of the complex Ginzburg-Landau equation with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (a) Model score vs noise intensity using the differential and weak forms of SINDyCP with nt = 4 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (b) Model score vs num- ber of samples for varying number of randomly generated tra- jectories, varying trajectory length, and noise intensity 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' generated, with (b, c) distributed as Gaussian random variables with means (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0) and standard deviations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For too little data, the fit fails to identify the correct model, and the value of 1 − R2 is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The models improve moderately with an increasing number of samples per trajectory (the product of Nt with the number of spatial grid points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' More importantly, a suf- ficiently large number of trajectories nt is required to achieve a good fit (at least 3 in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' These requirements should be carefully assessed in order to achieve successful SINDyCP fits for more general pattern forming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Parameter extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='—As a final demonstration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' However, in typical pattern formation applications, one does not have direct control over such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Extrapolation of localized states in the cubic-quintic Swift-Hohenberg equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Red dotted lines show the values used to train the SINDyCP fit, and dashed colored lines show the coefficients derived from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (b) Localized states extrapolated from numerical simulations of the SINDyCP fit with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, corresponding to the black dotted line in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' tedious calculations to derive, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=', an expansion and cen- ter manifold transformation around a bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The SINDyCP approach enables an automated discovery of such relationships, which can be used to extrapolate system behavior beyond a set of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We perform the weak SINDyCP fit using these trajectories subject to 1% injected noise with a quadradic parameter library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Remarkably, even with limited and noisy training data, the method identifies an accurate relationship between ε and the normal-form parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Thus, simulations of the identified model with random initial conditions converge to localized states [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4(b)] for ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1 despite the significant extrapolation of the parameter value beyond the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Discussion—The SINDyCP approach represents a simple but powerful generalization of SINDy with con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This work benefited from insightful discussions with Alan Kaptanoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Zachary G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 111, 13004-13009 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [41] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Case, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Wee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Driscoll, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Motter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Heterogeneity-stabilized homogeneous states in driven media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 12, 4486 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 1 Supplementary Material for “Data-driven discovery and extrapolation of parameterized pattern-forming dynamics” Zachary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nicolaou, Guanyu Huo,Yihui Chen, Steven L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Brunton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Nathan Kutz S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' NUMERICAL INTEGRATION For the complex Ginzburg-Landau equation, we use a pseudospectral integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The mode amplitudes are determined by σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01 for the four trajectories used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We also provide an animation showing the phase and amplitude for longer runs of 100 time units (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0/(1 + � |n|)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Snapshot of the animation showing the phase φ and amplitude r of the trajectories, where A = reiφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 0 r/2 3π/2 2rl L/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='2 0 L/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='9 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6 r L L/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0 L/2 L L/2 0 L/2 7-7 L/2 0 L/2 7 x x2 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' DEMONSTRATIONS Demonstrations of SINDyCP in discrete maps, ODEs and PDEs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This equation is the model for a universal period-doubling route to chaos as the parameter r increases past 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='56995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We perform the SINDyCP fit using four sample trajectories of 1000 iterations, corresponding to parameter values r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='9 (red dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' This equation exhibits the iconic butterfly-shaped Lorenz attractor for certain parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We perform the SINDyCP fit using five sample trajectories that have converged to their attractors, corresponding to the randomly selected parameter values σ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='0, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='9, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5, ρ = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='2, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, and β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Finally, the right panels illustrate the CGLE described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' SINDyCP ft Input data Model Logistic map Lorenz system Complex Ginzburg-Landau equation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Demonstrations of the SINDyCP approach for three models (top row) of nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 3 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' OREGONATOR MODEL AND NORMAL FORM TRANSFORMATION We mainly follow the analyses of the Oregonator model in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [30,32], with realistic parameter values shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For small µ, the weakly nonlinear theory follows from a perturbative expansion of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Take x ≡ (CX, CY , CZ) − (C0 X, C0 Y , C0 Z) and express Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (4)-(6) as ˙x = F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Define the multilinear operators of partial derivatives Fxn(ei1, · · · , ein) = ∂nF/∂xi1 · · · ∂xin with ei the ith component unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We develop a transformation x = y+h(y, µ) perturbatively, where y ≡ Aeiω0tu + ¯Ae−iω0t¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2), where b ≡ Im(d)/Re(d) = b0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='173 and c ≡ −Im(g)/Re(g) = c0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' As expected, these parameter values correspond to the amplitude turbulence regime of the CGLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' For our numerical simulations, we use a spatial domain of length L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We strobe the time in steps of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='94804 s, which corresponds to the critical frequency of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We then interpolate the time series in steps of T/1000 to generate the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' The first 200 time steps are discarded as the trajectories relax to their attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 2(b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' k1 k2 k3 k4 k5 DX DY DZ CH CA CB/(1 − µ) ν 2 106 10 2 × 103 1 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='6 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='787 1 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Parameter values for the Oregonator model, in cgs units (suppressed for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 4 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' WEAK FORMULATION IMPLEMENTATION We refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' [25-28] for the theory of the weak formulation of SINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Here, we only briefly describe our efficient numerical integration method for the weak formulation used in pysindy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We suppose that the spatial grid is one-dimensional, for the moment, and the values of the coordinates on the grid points are xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (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))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (2(p − k) − d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='x2(p−k)−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We linearly interpolate the function as f(x) = fi + x−xi xi+1−xi (fi+1 − fi) where xi ≤ x ≤ xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' Φ(d)(x) = φ(d−1)(x) for d > 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' By relabelling the dummy summation variables, we can recast Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S13) 5 Expressing the integrals along each dimension as dot products [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S10)] enables efficient vectorization with BLAS operations, and the integration weights [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' (S11)-(S13)] only need to be evaluated a single time when the library is first initialized (in a vectorized fashion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} +page_content=' We further vectorize the code by forming tensor products over all integration dimensions to calculate multidimensional integrals using a single tensor dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE0T4oBgHgl3EQfzgJq/content/2301.02673v1.pdf'} diff --git a/I9AzT4oBgHgl3EQfVPxY/content/tmp_files/load_file.txt b/I9AzT4oBgHgl3EQfVPxY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f6825dea3d64f8576bd79e0b56bd57321e582b6 --- /dev/null +++ b/I9AzT4oBgHgl3EQfVPxY/content/tmp_files/load_file.txt @@ -0,0 +1,156 @@ +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'} 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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'} diff --git a/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf b/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf new file mode 100644 index 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sha256:009ca451a59af2c8a8fde5e6dea031b36ce7f1c12d49a881e49877f8d3ba012a +size 170996 diff --git a/KNE3T4oBgHgl3EQfXwo4/content/tmp_files/2301.04481v1.pdf.txt b/KNE3T4oBgHgl3EQfXwo4/content/tmp_files/2301.04481v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19e69aa93a7f8f922247f2d6faab6e75ed12d537 --- /dev/null +++ b/KNE3T4oBgHgl3EQfXwo4/content/tmp_files/2301.04481v1.pdf.txt @@ -0,0 +1,1513 @@ +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) + diff --git a/KNE3T4oBgHgl3EQfXwo4/content/tmp_files/load_file.txt b/KNE3T4oBgHgl3EQfXwo4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..48c345500992ae82053b9565112ec4e298bbe1f5 --- /dev/null +++ b/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.' 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100644 index 0000000000000000000000000000000000000000..2b19b49ef6c36d60bfdabe7c1965993c587dce39 --- /dev/null +++ b/MtAzT4oBgHgl3EQfkv1s/content/tmp_files/2301.01536v1.pdf.txt @@ -0,0 +1,671 @@ +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 +1. Lines M. E., Glass A. M., Principles and Applications of Ferroelectric and Related Materials, Clarendon, Oxford, +1977. +2. Samara G. A., Ferroelectrics, 1973, 5, 25, doi:10.1080/00150197308235776. +3. Blinc R., J. Phys. Chem. Solids, 1960, 13, 204, doi:10.1016/0022-3697(60)90003-2. +4. McMahon M. I., Nelmes R. J., Kuhst W. F., Dorwarth R., Piltz R. O., Tun Z., Nature, 1990, 348, 317, +doi:10.1038/348317a0. +5. Nelmes R. J., McMahon M. I., Piltz R. O., Wright N. G., Ferroelectrics, 1991, 124, 355, +doi:10.1080/00150199108209465. +6. Robertson J. M., Ubbelohde A. R. J. P., Proc. R. Soc. London, Ser. A, 1939, 170, 222, +doi:10.1098/rspa.1939.0028. +7. Reiter G. F., Mayers J., Platzman P., Phys. Rev. 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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 + diff --git a/MtAzT4oBgHgl3EQfkv1s/content/tmp_files/load_file.txt b/MtAzT4oBgHgl3EQfkv1s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a62b7aced32231b69f196114e4b26732cd7b0928 --- /dev/null +++ b/MtAzT4oBgHgl3EQfkv1s/content/tmp_files/load_file.txt @@ -0,0 +1,796 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf,len=795 +page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 4, 43708: 1–12 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='43708 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='ua/journal Path integral Monte Carlo simulations of the geometrical effects in KDP crystals F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Torresi , J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Our model results show the Ubbelohde or geometrical effect (GE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', the expansion of the H-bond with deute- ration, in agreement to what is observed in H-bonded ferroelectrics with short H-bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Key words: ferroelectric phase transition, H-bonded ferroelectrics, path integral Monte Carlo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Besides the technological interest, KDP has also attracted much attention due to its rich, complex and intriguing phenomenology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', the huge isotope effect that displays associated to its ferroelectric-paraelectric (FE-PE) phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' With deuteration, the critical temperature 𝑇𝑐 changes from ≈ 122 K to ≈ 210 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The saturated polarization 𝑃𝑠 at low 𝑇 also shows a large isotope effect, increasing from ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='0 µC/cm2 for KDP to ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='2 µC/cm2 for a sample with 98% of deuteration [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The origin of these strong isotope effects is still controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Ab initio calculations have recently shown that tunneling and geometric effects are complementary aspects of the same phenomenon[9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 43708-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='01536v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='mtrl-sci] 4 Jan 2023 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Torresi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Koval the H-bond geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' On the other hand, these results are in striking contrast with the very weak dependence obtained at fixed potential and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Thus, the proton tunneling subunit and the host lattice are strongly coupled and the host-and-tunneling system is not separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Many models were successfully developed in the past to shed light into the general phenomenology of H-bonded ferroelectric materials [11–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' First, we develop a three-site model for the H-bond to study local quantum geometric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The model parameters are fitted using recent ab initio results [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The paper is organized as follows: in the next section we explain the models used and describe details of the PIMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Section 3 describes and discusses the results obtained for the three-site model and for the linear chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Finally, we elaborate a summary and our conclussions in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Models and calculation details 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Three-site model ��� ��� ��� ��� � � Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (Colour online) H-bond parameters in the three-site model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 𝑅 ≡ 𝑅OO is the distance between oxygen nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 𝑟OH is the proton-oxygen distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The variable 𝛿 = 𝑅OO −2𝑟OH is defined as the distance between the two possible equilibrium positions of the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Then, 𝑥 = 𝑅OO/2 − 𝑟OH is the proton coordinate relative to the H-bond center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This parameter definition is also used in the linear chain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' With the aim to model linear H-bonds, a Double Morse (or back-to-back) potential (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='1) where 𝑅 is the O–O distance, and 𝑥 represents the H position relative to the H-bridge center (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' However, simulations at low temperature with the potential described in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='1, relaxing both variables 𝑥 and 𝑅, yield to a collapse of the potential barrier and the equilibrium energy profile displays one minimum only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Therefore, it is mandatory to introduce a new interaction which preserves the system from the O–O distance collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This O–O potential will represent the interaction between both oxygens and the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The following Morse potential between oxygens is chosen [19]: 𝑉OO (𝑅) = 𝐷OO � 1 − e−𝑎OO(𝑅−𝑅0)�2 − 𝐷OO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Moreover, 𝑅 diminishes with decreasing 𝑥, which is the inverse situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This 1D linear model consists of a chain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='O–H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='O–H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='O–H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='O–H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' There are two atoms, one oxygen and one hydrogen in each unit cell (O–H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The supercell of dimension 𝐿 = 200𝑅 is subjected to periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' In the simulation, 𝐿 is allowed to relax at zero stress, as well as each coordinate 𝑥𝑖 and 𝑅𝑖 of each unit cell 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3), and the last term is the short-range interaction energy between protons/deuterons 43708-3 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Torresi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Koval Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (Colour online) Schematic representation of the 1D chain model in the ordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Each unit cell is formed with one oxygen (red sphere) and one hydrogen (white sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Our model consists of a supercell subjected to periodic boundary conditions containing 200 unit cells (for better visualization only 8 unit cells are shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' stemming from the ice rules restrictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', in this 1D model, only one proton is attached to each oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The last sum in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='4 is restricted to nearest neighbours for each index ⟨𝑖𝑗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' There is no long-range part in this model, which precludes a phase transition in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The effective action in this case allows us to signifi- cantly reduce the Trotter number 𝑀 required for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' In all the simulations performed we have used 𝑀 = 128 beads for the quantum polymer associated with each atom in the O–H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='O bonds, which yielded well-converged results [19, 25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Additionally, a normal-mode representation of the quantum polymers was used in order to ensure ergodicity in the MC sampling [28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The simulations for the 3S model consisted of 1 × 106 MC steps preceded by 5 × 105 steps of thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' In the 1D chain model simulations, we took 3 × 104 steps of thermalization plus 1 × 105 MC steps for computing averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' In this case, each calculation performed was an average of 20 runs with different random number generator seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The centroid is defined as the center of mass (CM) of the polymer and represents the average position of the quantum particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Results and discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Geometrical effect study using the three-site model The six potential parameters of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3) have been fitted in order to perform the GE study with the 3S model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' First, we fixed the values of 𝑎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='89 Å −1 [20, 21] and 𝐷 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='12 eV of the model parameters for the proton potential defined in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (Colour online) Proton/Deuteron probability distribution contours for the three-site PIMC simulations at 𝑇 = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' different H-bonded compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' There is a strong correlation between the OH and OO distances for the family of H-bonded compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The equilibrium distance 𝑟OH diminishes systematically with increasing 𝑅 for 𝑅 > 𝑅𝑐 [33, 34], reaching a saturated value around 𝑟∞ OH ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='95 Å for very large 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Therefore, we took the parameter value 𝑟0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='93 Å so that the values 𝑥 that minimize 𝑉OH (𝑥, 𝑅) in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='1) with the OO distance 𝑅 fixed [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' On the other hand, the parameter values for the OO interaction 𝑉OO (𝑅) [see equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='2)], were initially taken from reference [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='55 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='6 Å [36], with 𝑅cl OO lying precisely in the middle of that window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The final parameter values for the 3S model are shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Potential parameters used in the 3S model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 𝐷 [eV] 𝑎 [ Å−1] 𝑟0 [ Å] 𝐷OO [eV] 𝑎OO [ Å−1] 𝑅0 [ Å] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The OO distance for the peak positions are in each case: 𝑅peak OO (𝐻) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='527 Å and 𝑅peak OO (𝐷) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='543 Å, which represents a distance enlargement for the OO bond of Δ𝑅OO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='016 Å, evidencing the geometrical or Ubbelohde effect of the H-bond expansion with deuteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Moreover, the corresponding average values also increase with 43708-5 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Torresi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (Colour online) Distribution of the radius of gyration 𝑟𝐺 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' centroid coordinate 𝑥𝐶𝑀 for the three-site simulations at 𝑇 = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' deuteration: ⟨𝑅OO(𝐻)⟩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='525 Å and ⟨𝑅OO(𝐷)⟩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='540 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The PIMC simulations also show a change in the variable 𝛿 with deuteration for the peaks observed in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The variation is: Δ𝛿 = 𝛿𝐷 − 𝛿𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='079 Å, where 𝛿𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='417 Å and 𝛿𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='496 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This is also reflected in a shrinking of the O–H bonds: Δ𝑟 = 𝑟OH − 𝑟OD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='032 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Thus, our simple 3S model satisfactorily reproduces the isotopic geometrical effects for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' It is worth to notice that if the OO distance is not allowed to relax, then the GE is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' For instance, we have fixed the value 𝑅OO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='021 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Comparing this result with that considering the oxygen dynamics (Δ𝑟 = 𝑟OH − 𝑟OD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='032 Å), we observe an increment of ≈ 50% in the isotopic geometrical effect in the case where the oxygens are allowed to relax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' For the deuteron, being less delocalized than the proton, the selfconsistent effect is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', the extension of the quantum polymer) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Notice that the largest values of 𝑟𝐺 are found at 𝑥𝐶𝑀 ≈ 0 where delocalization is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This yields the isotopic geometrical effect, which is observed in the calculated probability distribution of figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Now, we focus on the development of a 1D chain model, described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='2 [see equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='5)], which was specifically designed to explain the isotope effects in the phase transition of KDP and was fitted to ab initio results [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' This more realistic 1D model has, in the classical nuclei version, a ferroelectric-paraelectric transition at 𝑇 ≈ 350 K [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' We start from equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='5 for the 1D model, which has seven parameters to be adjusted for the KDP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3), have been adjusted to reproduce six magnitudes obtained from ab initio calculations for KDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Finally, we have fitted the remaining parameter 𝐽 that corresponds to the proton-proton interaction term in equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' To this end, 𝐽 was adjusted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The final values for the parameters used in the 1D model are listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Potential parameters used in the 1D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 𝐷 [eV] 𝑎 [Å−1] 𝑟0 [Å] 𝐷OO [eV] 𝑎OO [Å−1] 𝑅0 [Å] 𝐽 [ eV/Å 2] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='838 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='966 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='831 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3𝑚 𝑝 and 𝜇𝐷 = 3𝑚 𝑝, respectively, with 𝑚 𝑝 the proton mass [9, 10, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Torresi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Koval (a) Proton (b) Deuteron Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' (Colour online) Proton/Deuteron probability distribution contours in the H-bonds for the linear- chain PIMC simulation at 𝑇 = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 𝑇𝑐 in the FE phase of KDP [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', the expansion of the H-bonds in the chain with deuteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The O–O distance for the peak positions are in each case: 𝑅peak OO (𝐻) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='515 Å and 𝑅peak OO (𝐷) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='542 Å, which represents a distance enlargement for the O–O bond of Δ𝑅OO ≡ 𝑅OO(𝐷) − 𝑅OO(𝐻) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='027 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The 𝑥 coordinate of the peak position also expands with deuteration, from 𝑥peak 𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='188 Å to 𝑥peak 𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='218 Å, with a net increase of Δ𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='030 Å or similarly Δ𝛿 ≡ 𝛿𝐷 − 𝛿𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='060 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' These results are summarized in table 3 and compared with the available experimental data for KDP and DKDP [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Nuclear quantum calculations of the H-bond geometries for KDP and DKDP using the 1D linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The results, which correspond to the peak positions of figure 5, are contrasted with the experimental data of reference [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Distances are in Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' PIMC KDP (𝜇𝐻 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3 𝑚 𝑝) DKDP (𝜇𝐷 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='0 𝑚 𝑝) Δ𝑅OO Δ𝛿 results 𝑅OO 𝛿 𝑅OO 𝛿 1D model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='376 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='060 Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' [40] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='385 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Distribution of the radius of gyration 𝑟𝐺 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' the potential barrier, to a lower bound of 𝑅min ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='3 Å near 𝑥 = 0 as shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Thus, we conclude that tunneling is assisted by the 𝑅 distance modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' In other words, the tunneling for the deuteron is very scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' These results are consistent with the general assumption in the tunneling model: protons are capable of tunelling while deuterons are not [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The model was fitted to reproduce general features for a large set of different H-bonded compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The results show an increase with deuteration of 𝑅 and 𝑥 for the observed peaks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', a GE, which is in agreement with that observed in H-bonded compounds with strong H-bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' During the PIMC simulations we have also plotted the instantaneous radius of gyration 𝑟𝐺 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' the centroid position 𝑥𝐶𝑀 of the quantum polymer representing the proton/deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Torresi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Koval more localized in both sides and far from the O–H–O bond center, with small values of 𝑟𝐺 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', a more classsical behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' These features yield a more effective contraction of the O–O bond in the proton case, explaining the GE observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The parameters of the 1D model were fitted to ab initio results for KDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The calculated probability distribution contours show only one peak in the (𝑅, 𝑥) configuration space for both proton/deuteron cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The distribution is more asymmetric in the proton case due to the anharmonicity of the potential and the quantum delocalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The distribution pattern is in qualitative agreement with the experimental distribution determined by high- resolution neutron diffraction studies [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The probability distribution contours show a peak which shifts substantially with deuteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' The distribution of the radius of gyration vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' We have shown that proton tunneling is assisted by a strong contraction of the O–O distance in the 1D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Acknowledgements We acknowledge support from Consejo Nacional de Investigaciones Científicas y Técnicas (CON- ICET), Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lines M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=', Glass A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Lasave, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Koval Метод iнтегралiв за траєкторiями у моделюваннi Монте-Карло геометричних ефектiв у кристалах KDP Ф.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Торрезi, Х.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Ласаве, С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Коваль Iнститут фiзики Росарiо, Нацiональний унiверситет Росарiо та Нацiональна рада з науково-технiчних дослiджень, вул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' параметри моделi вибрано так, щоб пояснити рiзноманiтнi загальнi влас- тивостi низки сполук з водневими зв’язками.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Використовуючи для параметрiв потенцiалiв результати першопринципних розрахункiв, розвинено одновимiрну модель, в якiй бiлiнiйнi протон-протоннi взаємодiї розглядаються в наближеннi середнього поля.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Ця модель вико- ристовується для дослiдження квантових ефектiв у ядрах, якi призводять до виникнення геометричного ефекту в кристалах KDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Пiдхiд IТМК дає змогу виявити, що протони тунелюють бiльш ефективно вздовж одновимiрного ланцюжка, нiж дейтрони;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' це спричиняє появу сильного притягувального центра, який зменшує вiдстань мiж атомами киснiв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} +page_content=' Ключовi слова: сегнетоелектричний фазовий перехiд, сегнетоелектрики з водневими зв’язками, метод iнтегралiв за траєкторiями у моделюваннi Монте-Карло 43708-12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfkv1s/content/2301.01536v1.pdf'} diff --git a/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf b/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..11e29a2ba94fe4c6d260dbb92576d77017218136 Binary files /dev/null and b/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf differ diff --git a/MtE2T4oBgHgl3EQfqgg6/content/tmp_files/2301.04039v1.pdf.txt b/MtE2T4oBgHgl3EQfqgg6/content/tmp_files/2301.04039v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba13d5cfd37f4f044dc668ec77601bc446ab1a63 --- /dev/null +++ b/MtE2T4oBgHgl3EQfqgg6/content/tmp_files/2301.04039v1.pdf.txt @@ -0,0 +1,232 @@ +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 + diff --git a/MtE2T4oBgHgl3EQfqgg6/content/tmp_files/load_file.txt b/MtE2T4oBgHgl3EQfqgg6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef8624b406cd0c9dcaf25fe86ef957a9c06cfbed --- /dev/null +++ b/MtE2T4oBgHgl3EQfqgg6/content/tmp_files/load_file.txt @@ -0,0 +1,114 @@ +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'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'} +page_content=' Weinberg, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'} +page_content=' Phys.' metadata={'source': 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+Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +Effects of turbulent diffusion and back-reaction on the dust distribution +around two resonant planets +Francesco Marzari1★ and Gennaro D’Angelo,2† +1Department of Physics and Astronomy, University of Padova, via Marzolo 8, I-35131, Padova, Italy +2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA +Accepted 1 February 2023. Received 1 February 2023; in original form 1 February 2023 +ABSTRACT +In evolved and dusty circumstellar discs, two planets with masses comparable to Jupiter and Saturn that migrate outwards while +maintaining an orbital resonance can produce distinctive features in the dust distribution. Dust accumulates at the outer edge of +the common gas gap, which behaves as a dust trap, where the local dust concentration is significantly enhanced by the planets’ +outward motion. Concurrently, an expanding cavity forms in the dust distribution inside the planets’ orbits, because dust does +not filter through the common gaseous gap and grain depletion in the region continues via inward drifting. There is no cavity +in the gas distribution because gas can filter through the gap, although ongoing gas accretion on the planets can reduce the gas +density in the inner disc. Such behaviour was demonstrated by means of simulations neglecting the effects of dust diffusion due +to turbulence and of dust backreaction on the gas. Both effects may alter the formation of the dust peak at the gap outer edge and +of the inner dust cavity, by letting grains filter through the dust trap. We performed high resolution hydrodynamical simulations +of the coupled evolution of gas and dust species, the latter treated as pressureless fluids, in the presence of two giant planets. We +show that diffusion and backreaction can change some morphological aspects of the dust distribution but do not alter some main +features, such as the outer peak and the expanding inner cavity. These findings are confirmed for different parametrizations of +gas viscosity. +Key words: accretion, accretion discs — methods: numerical — planets and satellites: gaseous planets — planet–disc interactions +1 INTRODUCTION +In a recent study, Marzari et al. (2019) examined the distribution of +dust particles around two resonant planets embedded in a circum- +stellar disc. The two planets, with masses comparable to those of +Jupiter and Saturn, had orbits in a resonant configuration, with ratios +of the mean motion equal to either 2:1 or 3:2. Because of the type +of resonance and of the applied disc conditions, the planets tend to +migrate outwards and dust particles tend to accumulate outside of +the orbit of the exterior planet. Concurrently, the inward migration +of dust grains that move inside of the orbit of the interior planet +leads to an enlargement of the dust gap compared to the gap in the +gas and to a dynamical decoupling between the gaps in the gas and +dust distributions. The build-up of the dust density at the outer edge +of the gap surrounding the planets is markedly higher in the case +of the 2:1 mean-motion resonance and may appear as a bright ring +(at appropriate wavelengths) in resolved observations of discs. A +similar phenomenon was also found for lower-mass planets (in the +Super-Earth mass range, Marzari & D’Angelo 2020), although less +pronounced. All those simulations were performed without includ- +ing the effects of possible dust diffusion due to gas turbulence and +of dust back-reaction on the gas. Here we explore the consequences +★ E-mail: francesco.marzari@pd.infn.it +† E-mail: gennaro@lanl.gov +of these two mechanism on the accumulation of dust grains at the +outer edge of the gap and on the formation of a wider gap in the dust +distribution compared to the density gap in the gas. This latter feature +may lead to the formation of a transition disc, if the planets are close +enough to the star (low gas surface density inside the planets’ orbits +can be caused by ongoing accretion of gas on star and planets). +Circumstellar discs are likely turbulent (e.g., Hughes et al. 2011). +Various mechanisms have been proposed as drivers of turbulence, +such as convective instability (e.g., Klahr & Hubbard 2014; Lyra +2014), vertical shear instability (e.g., Urpin 2003; Nelson et al. 2013; +Stoll et al. 2017), and magneto-rotational instability (e.g., Balbus & +Hawley 1991, 1998, 2003). Gas turbulence may force dust grains to +diffuse (over a length-scale dictated by the type of turbulence), a pro- +cess that not only may affect dust accumulation but can also hinder +the efficiency of dust entrapment at radial location of gas pressure +maxima. In fact, according to Sierra et al. (2019) and Pinilla et al. +(2020), dust diffusion may reduce, or even prevent, significant con- +centrations of grains at locations of gas “bumps”, since the grains can +disperse out of the region. This process might affect the conclusions +of our previous results on the accumulation of dust at the outer edge +of the gaseous gap of two planets in resonance by letting dust seep +through the gap and reach the inner disc regions. If the effect is large +enough, the concentration of dust at those radial locations, obtained +in the numerical simulations by Marzari et al. (2019) and Marzari & +D’Angelo (2020), may be severely depleted or even largely absent. +© 2023 The Authors +arXiv:2301.13489v1 [astro-ph.EP] 31 Jan 2023 + +2 +Marzari and D’Angelo +In addition to dust diffusion, the back-reaction of dust on gas can +also impact the formation of grain concentration at a local pressure +maximum. According to Taki et al. (2016), dust back-reaction can +deform the pressure gradient of the gas when high-enough values +of the dust-to-gas mass ratio are reached. This may be the case of +the dust concentration attained at the outer edge of gaseous gaps, +observed in the simulations of two planets in resonance migrating +away from the star. +To test the relevance of these two mechanisms, diffusion and back- +reaction, on the formation of dust over-dense regions caused by the +outward migration of two planets in resonance, we performed simula- +tions of the evolution of two planets in resonance in which both these +two mechanisms are taken into account. In Section 2, we describe +the numerical model exploited to study the coupled evolution of dust +and gas in presence of the two resonant planets. In Section 3, we +outline the dust behaviour when the planets evolve in the 3:2 mean- +motion resonance whereas, in Section 4, we analyse the case of the +2:1 mean-motion resonance. In Section 5, we test the robustness of +our results by changing the viscosity parameterization, including a +constant kinematic viscosity the one that applies a constant value of +the 𝛼 parameter. Finally, in Section 6, we discuss our results. +2 METHODS AND ALGORITHMS +In past work on the coupled evolution of dust and gas in proto- +planetary discs, we adopted a Lagrangian description of the solid +component. Instead, an Eulerian formalism is applied in the present +study, since solids are treated as pressureless fluids. Some details +on the involved equations are provided below to highlight the differ- +ences between the two approaches. Marzari et al. (2019) and Marzari +& D’Angelo (2020) used the two-dimensional (2D) FARGO hydro- +dynamics code (Masset 2000), modified to include the dynamical +evolution of dust particles embedded in the gaseous disc in a La- +grangian fashion (Picogna & Kley 2015; Picogna et al. 2018). +The drag force on the particles was computed from the local gas +density according to the equation (Woitke & Helling 2003) +F = +� +3𝐾 +3𝐾 + 1 +�2 +FE + +� +1 +3𝐾 + 1 +�2 +FS, +(1) +where FE is the Epstein drag contribution, given by +FE = −4 +3 +� +1 + 9𝜋 +128 𝑀2 +�1/2 +𝑠2𝜌𝑔𝑣thvrel, +(2) +and FS is the Stokes drag component, given by +FS = −1 +2𝐶𝐷𝜋𝑠2𝜌𝑔𝑣relvrel. +(3) +In the above equations, 𝜌𝑔 is the gas density, 𝑠 is the radius of the +particle, 𝑣th is the local thermal velocity of the gas and vrel is the +velocity of the dust particle relative to the gas. The quantity 𝐾 is the +Knudsen number and 𝑀 is the Mach number (computed from the +particle’s relative velocity 𝑣rel). Quantity 𝐶𝐷 is the drag coefficient +for the Stokes regime. +In this paper, to test the relevance of diffusion and back-reaction +on the formation of dust over-dense regions caused by the outward +migration of two planets in resonance, we carry out simulations with +the code FARGO3D (Benítez-Llambay et al. 2019). In this version +of the code, the dust particles are treated as additional pressureless +fluids where momentum is transferred between the gas and each +of the dust species (but not among dust species). The dust fluid is +affected by Epstein drag, which imparts a force per unit volume to a +dust species given by +F𝑑 = −𝜌𝑑 +Ω +𝜏𝑠 +(v𝑑 − v𝑔), +(4) +where 𝜌𝑑 is the dust density, 𝜏𝑠 is the Stokes number of the particle +and Ω is the Keplerian frequency of the gas. An equal and opposite +force is imparted to the gas. Note that, in Equation (4), information +regarding the drag coefficient are incorporated into 𝜏𝑠. +A term is added to the continuity equation to model the diffusion +of the dust species within the gas (Morfill & Voelk 1984) +𝜕𝜌𝑑 +𝜕𝑡 += ∇ · +� +𝐷𝑑𝜌∇ 𝜌𝑑 +𝜌 +� +, +(5) +where 𝜌 = 𝜌𝑔 + 𝜌𝑑 and 𝜌𝑑 is the density of individual dust species. +Equation (5) is only applied to the pressureless fluids and it assumes +the same diffusion coefficient for all dust species, which is taken +equal to the value of the gas kinematic viscosity (Brauer et al. 2008) +𝐷𝑑 = 𝜈. +(6) +The effects of this choice are not tested. +The original version of the code applies Stokes numbers, 𝜏𝑠, in +Equation (4) that are constants. The code was modified so that we +can select each pressureless fluid (dust species) not according to a +Stokes number, which varies as a function of the local properties of +the gas (density, temperature and velocity), but rather according to +the particle size. +3 DUST DISTRIBUTION NEAR PLANETS IN THE 3:2 +RESONANCE +We first investigate the case of a pair of planets that become captured +in the 3:2 mean-motion resonance and migrate outward thereafter. +The interior planet has the mass of Jupiter whereas the exterior +planet has the mass of Saturn. A more massive inner planet is a +condition conducive to outward migration. The planets orbit in a +cold, local-isothermal disc with a fixed aspect ratio 𝐻/𝑟 = ℎ = 0.02. +(Calculations with a larger ratio, ℎ = 0.05, are also presented.) +The disc extends from 0.4 to 12 au and is discretised over a grid of +512 × 1024 area elements (in the radius and azimuth, respectively). +The initial surface density of the gas declines as +Σ(𝑟) = Σ0 +�𝑟0 +𝑟 +� +, +(7) +where Σ0 = 200 g cm−2 is the density at the reference radius 𝑟0 = +1 au. +Three different populations of icy grains (bulk density 1 g cm−3) +are included in the simulations, whose sizes are 100 𝜇m, 1 mm and +1 cm. For the applied disc conditions, these particles have Stokes +numbers less than ≈ 0.1. +The initial dust-to-gas mass ratio for each of the three dust species +is 0.0033, so that the overall dust-to-gas mass ratio adds up to 0.01, +which is a typical value adopted for circumstellar discs and is based +on values found in the interstellar medium. However, dust needs not +be primordial in origin, that is, part of the inventory of solids from +which the planets formed. In fact, the dust populations may represent, +or contain, second generation dust produced by collisions among left- +over planetesimals, after the planets became massive enough (Turrini +et al. 2019; D’Angelo & Marzari 2022; Marta Bernabò et al. 2022). +The equations describing the evolution of the system are solved in +a non-inertial reference frame centered on the star, including the +indirect terms arising from the planets’ and disc’s gravity. +MNRAS 000, 1–9 (2023) + +Diffusion and back-reaction effects on dust +3 +4.00 +5.00 +6.00 +7.00 +8.00 + 0 + 20 + 40 + 60 + 80 + 100 + 120 + a (au) +Time (Kyr) +ν =1.e-6 +ν =5.e-6 +ν =1.e-5 + 0 + 0.04 + 0.08 + 0.12 + 0.16 + 0 + 20 + 40 + 60 + 80 + 100 + 120 +Eccentricity +Time (Kyr) +ν =1.e-6 +ν =5.e-6 +ν =1.e-5 +Figure 1. The top panel shows the semi-major axis of pairs of planets during +their migration, for three values of a constant kinematic viscosity, 𝜈, of the gas +(in units of 𝑟2 +0 Ω0, see text). In these cases, the planet pair is locked in the 3:2 +mean-motion resonance. In the bottom panel illustrates the evolution of the +orbital eccentricity. The inner, more massive planet has a lower eccentricity in +all cases. Data are averaged over a 250 yr window; see text for further details. +Various values of the gas kinematic viscosity, 𝜈, are considered be- +cause this parameter affects the tidal interactions between the planets +and the gas, and also determines the amount of dust diffusion through +Equation (6). Additionally, it can also affect the speed of the planets’ +outward migration. In these models, we adopt a constant value of +kinematic viscosity. The impact of 𝜈 on the efficiency of the outward +migration is illustrated in Figure 1, for a given value of the initial gas +density at the reference radius 𝑟0. In these models, the planets start to +migrate at the beginning of the simulations, when the distributions of +gas and dust are unperturbed (hence the initial rapid inward migra- +tion of the planets). As the outer planet approaches the inner planet +and their orbits become caught in resonance, the tidal perturbations +exerted by the outer planet on the circumstellar material alter the +torque balance on the inner planet. Consequently, the inner planet +first slows down and then migrates away from the star, pushing the +outer planet outward through the resonance forcing. +The evolution of the semi-major axis of the outer planet, 𝑎2, +is shown for three different values of the gas kinematic viscosity: +𝜈 = 10−6, 5 × 10−6, and 10−5, in units of 𝑟2 +0Ω0 (Ω0 is the Ke- +plerian frequency at 𝑟0). Note that a constant kinematic viscosity +corresponds to a variable 𝛼 parameter, 𝛼 ∝ 𝜈/(ℎ2√𝑟). With our +choice of parameters, at 𝑟 = 𝑟0, said values of 𝜈 would correspond to +𝛼0 = 0.0025, 0.0125, and 0.025, respectively. However, in the disc +regions where the planets orbit, 𝛼 would be smaller by a factor of 2 +or more. +After a different behaviour at the beginning of the evolution, prior +to or shortly after the capture into the 3:2 orbital resonance (see +Figure 1), the planets undergo sustained outward migration locked in +mean-motion resonance. The migration speed of the pair is related +to 𝜈 and determined by the shape of the common (or overlapping) +gaseous gap of the two planets. A similar outcome is obtained for +the cases involving the 2:1 resonance, as shown in the next section. +Notice that the outer planet is subject to a negative torque exerted by +the disc material exterior to its orbit, hence it would tend to move +inward, whereas the resonance forcing pushes it outward. These two +opposing torques allow the resonance to be maintained during the +outward migration of the pair. +In the calculations presented herein, all disc material exerts torques +on the planets, including material within the planets’ Hill spheres. +Since the numerical resolution is limited and density variations close +to the planets may not be properly resolved, some spurious effects +may arise that affect outward migration. To quantify possible dif- +ferences, one model was also performed by removing the torques +exerted from within the planets’ Hill spheres. The orbital evolution is +comparable to that of the calculation with default setup, although at +a somewhat reduced outward migration speed. The amount of kine- +matic viscosity can also alter the local distribution of material around +the planets and unresolved density gradients can possibly impact the +resulting migration velocity. Nonetheless, it must be pointed out that +for the purposes of this study the details of the outward migration +process are not important, henceforth tests on the response of the +system to numerical parameters are unnecessary. The only require- +ment is that the planet pair becomes locked in resonance and moves +away from the star for a prolonged amount of time. +When the outer planet reaches 6.5 au from the star, we compare +the dust distributions in the three cases with different viscosity. This +comparison is shown in Figure 2. The dust density profiles show a +peak at the outer border of the gas gap carved by the planets’ tidal +perturbations. This peak is more marked in the distributions of the +largest grains, which are less coupled to the gas (i.e., they have a +larger Stokes number and therefore a longer coupling timescale). +Dust-to-gas mass ratios at the peak location range from 0.04 to 0.08, +increasing as 𝜈 decreases. In the region inside the inner edge of the +gap there is a significant depletion of dust due to drifting motion +towards the star. Re-supply of dust to this region is reduced, or +halted, by the dust trap at the outer edge of the gas gap. After some +time, the disc would appear as a transition disc with an inner cavity +in the dust density, which expands outward over time due to the +outward migration of the planets. These effects appear more evident +at lower gas viscosity, which may be due to the lower level of diffusion +but it may also be related to the different migration velocity of the +planet pair (see Figure 1), as discussed below. Beyond ≈ 10 au +the disc is depleted of mm- and cm-grains (but not of the smallest +grains), an effect associated to the inward drift of the particles via +gas drag (which does not affect as much the smallest grains). This +is a boundary effect related to the fact that particles are not flowing +inward from greater distances (i.e., there is not re-supply of solids at +the outer boundary). Test simulations not reported here confirm this +issue. +The accumulation efficiency of particles at the outer edge of the +gas gap depends on the stopping time of the particles (𝜏𝑠/Ω) and +the timescale over which the radial pressure gradient of the gas +moves (in our case, due to planet migration). For a given stopping +time, the shorter the outward migration timescale, the less time dust +grains have to accumulate. Stated differently, for a similar orbital +MNRAS 000, 1–9 (2023) + +4 +Marzari and D’Angelo +10-2 +10-1 +100 + 2 + 4 + 6 + 8 + 10 + 12 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +10-2 +10-1 +100 + 2 + 4 + 6 + 8 + 10 + 12 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +10-2 +10-1 +100 + 2 + 4 + 6 + 8 + 10 + 12 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +Figure 2. Surface density (averaged in azimuth around the star) of dust +particles of different sizes, ranging from 100 𝜇m to 1 cm. These profiles are +compared to the re-scaled gas density (i.e., multiplied by 0.0033; the total +dust-to-gas mass ratio is 0.01). The top panel refers to a kinematic viscosity +equal to 𝜈 = 10−6, the middle panel to 𝜈 = 5 × 10−6 and the bottom panel to +𝜈 = 10−5, in units of 𝑟2 +0 Ω0. +10-2 +10-1 +100 + 2 + 4 + 6 + 8 + 10 + 12 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +Figure 3. Dust and gas profiles (as in Figure 2) with 𝜈 = 10−5 𝑟2 +0 Ω0, but +without the inclusion of diffusion and back-reaction. +configuration of the planets, i.e., similar orbital frequencies Ω, a +more rapid outward migration can facilitate the filtering process of +dust toward the star. The overall outcome is a less depleted inner disc +in the cases with more vigorous outward migration. +For the same reason, the different migration speed also controls +the build up of the dust at the outer edge of the gas gap. As a +consequence, the reduction in the peak density at the outer edge of +the gap observed at higher viscosity values, shown in Figure 2, is +likely due to the combination of the two effects: a higher diffusion +rate and a decrease in the trapping efficiency of dust grains caused +by the faster outward migration rate of the planets. +To better characterize the role of diffusion and back-reaction in +the formation, size and shape of the outer peak in dust density, we +ran a simulation in which both these two processes were neglected, +adopting a kinematic viscosity 𝜈 = 10−5 𝑟2 +0Ω0 (see Figure 3). In +this model, the migration speed is broadly consistent with that of the +model illustrated in the bottom panel of Figure 2, but the peak in the +dust density distributions of the largest particles tend to be higher +and sharper compared to corresponding one in the the bottom panel +of Figure 2. The implication is that diffusion is indeed affecting the +concentration of the dust by spreading large grains over a broader +radial region. Nonetheless, the reduced efficiency in collecting dust +caused by diffusion is not strong enough to prevent the formation of +a prominent density peak outside the planets’ orbits, which may be +detected by high resolution observations. +In Figure 4 we plot the surface density distribution of the gas and +of dust grains of different sizes for the case with lowest viscosity, +𝜈 = 10−6 𝑟2 +0Ω0. The gas gap is significantly narrower than that of the +dust and the width of the latter is larger for larger grain size. For 1 cm +size dust grains, the density distribution is confined in an over-dense +ring at the outer edge of the gas gap. Therefore, according to these +results, dust diffusion and back-reaction are not able to prevent the +formation of narrow rings in the dust distribution at the outer edge +of the gas gap. In our simulations there is no re-supply of dust at +the outer boundary of the simulated disc, but it is expected that a +continuous distribution of solids beyond the grid boundary would +supply dust to the inner disc regions. In this case, at the outer edge +of the gap, we would observe an enhanced density, as predicted by +our simulations. Beyond this density peak, however, there would be a +continuous distribution of dust originating from more distant regions. +Over timescales much longer than those simulated here, dust drifting +MNRAS 000, 1–9 (2023) + +Diffusion and back-reaction effects on dust +5 +from larger distances may also affect the density peaks at the outer +edge of the gas gap. In fact, an additional simulation with a wider +radial boundary (not reported here) does show enhanced peaks in +large grains, due to solids drifting from farther distances. Also the +population of small grains would increase at the peaks over longer +times, but at a slower rate, dictated by the drift velocity. Nonetheless, +the depletion of dust within the inner edge of the gas gap would not +be affected by this process because the outer dust trap appears to be +efficient enough to halt (or severely impede) refilling of grains. In +fact, if refilling of the inner disc was sustained, it would occur within +the time of our simulations but it is not observed. +Continued supply of dust from farther out in the disc may, at some +point, raise the density in the peak regions beyond some threshold +value to make the peaks unstable. For example, the dust-to-gas mass +ratio may become large enough to induce a back-reaction response +on the gas that redistributes the particles over some radial region via +collisions and/or enhanced dust coagulation may ensue (coagulation +into larger particles would reduce the back-reaction force exerted +on the gas, because of the lower surface-to-mass ratio). The back- +reaction response may also smooth out the gas density gradient in +the radial direction, altering the radial pressure gradient and reducing +its ability to retain grains. Such processes are not considered in this +study. +To test the impact of a larger aspect ratio (i.e., warmer disc) on +the formation of the inner dust cavity and of the peak external to +the gas gap, we performed two additional simulations adopting ℎ = +0.05. In the first model we used a higher gas surface density (Σ0 = +800 g cm−2), in order to increase the speed of planet, migration while +in the second we used the same density as in the previous cases +(Σ0 = 200 g cm−2). In the latter simulation, since the rate of outward +migration is smaller, the planets are located closer to the star in the +plot. In the top panel of Figure 5, the density profiles are shown for +the different grain sizes in the high density case. The bottom panel +illustrates those profiles for the low density case. In both simulations +the density patterns are similar to those in Figure 2, suggesting that +a higher aspect ratio does not impair the ability of two planets in +resonance to carve an inner gap in the distribution of the larger +grains and to trap dust at the gap’s outer edge (dust-to-gas mass +ratios around the peak region are 0.02–0.03). Thus, dust filtering +through the outer edge of the gas gap is not increased by the different +morphology of the dust trap in these warmer discs. +4 DUST DISTRIBUTION NEAR PLANETS IN THE 2:1 +RESONANCE +To test the evolution of the dust when the planets are captured in a +2:1 mean-motion resonance, we decreased the gas density to Σ0 = +40 g/cm2 in order to induce orbital locking in this resonance. It is +known that capture in this mean-motion resonance is a more delicate +process than capture in the 3:2 resonance. If the relative migration +velocity with which the pair of planets approach each other is above a +certain threshold, the resonance forcing is overcome and convergent +migration continues. Additionally, once the 2:1 orbital resonance is +established, migration of the two planets typically proceeds inward +because of the way the two gas gaps overlap. In this configuration, +outward migration may be obtained by choosing an appropriately +low kinematic viscosity so that a wide, common gaseous gap forms +around the orbits of the two planets. +As in the the models of the previous section, the planets start to +migrate in unperturbed distributions of gas and therefore the initial +inward migration of the planets is artificially rapid. This choice af- +Figure 4. Surface density maps illustrating gas and dust distributions around +two planets locked in the 3:2 resonance (𝜈 = 10−6 𝑟2 +0 Ω0). The top panel +shows the gas density, whereas the second, third and fourth panels represent +100 𝜇m, 1 mm and 1 cm particles, respectively. +MNRAS 000, 1–9 (2023) + +9 +5 +102 +4 +azim (rad) +(g/cm 2) +3 +10 +M +2 +1 +0 +100 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)9 +5 +10 0 +4 +azim (rad) +(g/cm 2) +3 +10-1 +2 +1 +0 +10-2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)9 +5 +10 0 +4 +azim (rad) +3 +10-1 +M +2 +1 +0 +10-2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)6 +5 +100 +4 +azim (rad) +3 +10-1 +M +2 +1 +0 +10-2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)6 +Marzari and D’Angelo +10-1 +100 +101 + 2 + 4 + 6 + 8 + 10 + 12 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +10-2 +10-1 +100 + 2 + 4 + 6 + 8 + 10 + 12 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +Figure 5. +Dust and gas density profiles (as in Figure 2) in a disc with +𝜈 = 10−5 𝑟2 +0 Ω0 and aspect ratio 𝐻/𝑟 = ℎ = 0.05. In the top panel Σ0 = +800 g cm−2 whereas Σ0 = 200 g cm−2 in the bottom panel. +fects the radius at which the planets are trapped in resonance but is +otherwise not much relevant for the purposes of this study. As for +the models discussed in Section 3, the model with a larger value of +the kinematic viscosity results in a more vigorous outward migra- +tion of the planets once they become trapped in resonance. This is +illustrated in Figure 6 for cases with 𝜈 = 10−6 and 𝜈 = 10−7 𝑟2 +0Ω0 +(𝛼0 = 2.5 × 10−3 and 2.5 × 10−4, respectively) . Note, however, that +the orbits also become significantly eccentric, which also affects out- +ward migration (D’Angelo et al. 2006). We also tested a larger value, +𝜈 = 5 × 10−6 𝑟2 +0Ω0, but the outer planet crosses the 2:1 resonance +with the inner planet. The pair is temporarily trapped in the 5:3 res- +onance at which point it begins migrating outwards. That resonance +is then broken and the pair becomes finally captured in the 3:2 reso- +nance, continuing to migrate outward. For even higher values of the +kinematic viscosity, there is no trapping in the 2:1 resonance, which +might be attained by further reducing Σ0. However, this possibility +was not tested since it could lead to initial dust-to-gas mass ratios too +dissimilar from the other simulations presented herein. +The dust distributions are shown in Figure 7 for the two different +values of kinematic viscosity and, in the bottom panel, for a model +without the inclusion of diffusion and back-reaction. Because of +the significantly different migration speed, we only compare the dust +density distributions at the end of the simulation (when the planet are +6.25 +6.30 +6.35 +6.40 +6.45 +6.50 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 +a (au) +Time (Kyr) +ν =1.e-6 +ν =1.e-7 +0.00 +0.05 +0.10 +0.15 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 +Eccentricity +Time (Kyr) +ν =1.e-6 +ν =1.e-7 +Figure 6. The top panel shows the migration of the exterior planet of a pair, for +two values of a constant kinematic viscosity of the gas, 𝜈, as indicated in the +legend. The planet pair is locked in the 2:1 mean-motion resonance. Only the +semi-major axis of the exterior planet is plotted because of the larger mutual +distance compared, for example, to the 3:2 case. The orbital eccentricity of +both interior and exterior planets is illustrated in the bottom panel. Data are +averaged over a 250 yr window as in the case of the 3:2 resonance (Figure 1). +located at different orbital radii). For this resonance too a significant +dust enhancement develops at the outer edge of the gas gap, where +dust-to-gas mass ratios can achieve values of order unity. The inner +cavity is larger at all grain sizes, compared to that produced by the +3:2 resonant configuration. This is possibly related to the slower +outward migration rate of the planets in the 2:1 resonance, which +can reduce filtering of the dust through the orbits of the planets +and toward the star. Additionally, the lower gas density increases +the Stokes numbers of the grains, reducing their drift timescale and +facilitating grain removal from the inner disc. +Comparing the top and bottom panels of Figure 7, one can notice +significant differences in the dusty features exterior to the planets +orbits. Since the gas density is also different in the two models (at +those times), it is unclear how much of the difference is caused by +the action of diffusion and back-reaction of the solids. +The orbital eccentricity of the planets can drive asymmetries in +the distribution of the disc material. Consequently, both the gas gap +outer edge and dusty rings can become asymmetric (see Figure 8). +We did not perform a detailed analysis of ring asymmetries. However, +one possible explanation for the more asymmetric features arising +from the planets in the 2:1 resonance (compared to those in the 3:2 +resonance, see Figure 4), may be the larger eccentric perturbation +MNRAS 000, 1–9 (2023) + +Diffusion and back-reaction effects on dust +7 +driven by the inner (more massive) planet, which tends to have a larger +orbital eccentricity in the 2:1 resonance than in the 3:2 resonance (the +outer planets can have comparable eccentricities). +In Figure 8, the dust density distributions are shown for grains +of various sizes for the case with lowest viscosity, 𝜈 = 10−7 𝑟2 +0Ω0. +As mentioned, the width of the gap is significantly larger compared +to the case of the 3:2 resonance and the dusty ring at the outer +edge of the gas gap is evident for all particles, more than in the 3:2 +resonance. For the 2:1 resonance, the effects of the dust trap appear +more marked, both at the inner and at the outer edge of the gas gap. +It is expected that after sufficient time from the beginning of the +outward migration, the disc reduces to a single overdense ring at the +outer edge of the gap, close to the outer planet orbit. As discussed in +the previous section, this feature would result from the lack of dust +supply from larger orbital radii, beyond the outer boundary of the +grid. +5 COMPARISON WITH ALPHA VISCOSITY +To test the robustness of our results, we performed two additional +simulations, for the 2:1 resonance, in which the kinematic viscosity +is set as 𝜈 = 𝛼𝐻2Ω, where 𝐻 is the local pressure scale height of +the disc and the parameter 𝛼 is assumed to be a constant. Since we +work with a local-isothermal disc and a constant aspect ratio, 𝐻 ∝ 𝑟 +and 𝜈 ∝ 𝛼√𝑟. In the first model, we set 𝛼 = 10−3 whereas we set +𝛼 = 10−4 in the second. These parameters result in 𝜈 ≈ 10−6 and +≈ 10−7 𝑟2 +0Ω0 around the middle radius of the computational domain. +The outcomes of these simulations are illustrated in Figure 9, +after 4 × 104 yrs of evolution. The density peak at the outer edge +of the gas gap is clearly visible in both cases, although there are +differences in over-density morphology. In the top panel, at 𝛼 = +10−3, the peak appears similar at all sizes while, in the bottom panel +(𝛼 = 10−4), the peak is split in two for the largest particles (𝑠 = +1 mm and 𝑠 = 1 cm) and composed of three separate maxima for +𝑠 = 100 𝜇m. This behaviour was not observed for the 3:2 resonance +but it was already present in the simulation involving the 2:1 mean- +motion resonance obtained with a constant kinematic viscosity (see +Figure 7, top panel). A possible interpretation is that for the 2:1 +resonance multiple dust traps develop at the outer border of the gap. +By comparing the gas density distribution in the two cases shown in +Figure 9, this hypothesis appears to be confirmed since, in the case +with 𝛼 = 10−4, Σ at the outer border of the gas gap appears more +variable compared to that of the case with 𝛼 = 10−3. +The splitting of the peak in various maxima can be also observed +in the simulations of Marzari et al. (2019), for the case involving +a pair of planet locked in the 2:1 resonance, even if their physical +model (radiative disc), the initial parameters for the gas density and +the viscosity values are different. +6 CONCLUSIONS +According to the models presented by Marzari et al. (2019), two +planets in the mass range of Jupiter and Saturn, embedded in a +circumstellar disc and migratingoutwards,canstronglyaffect the dust +distribution in their surroundings. They can create a transition disc +with an inner cavity in the dust distribution that expands following +the outward migration of the planets. They can also build a strong +peak in the dust density at the outer edge of the gas gap carved +by the resonant planets, which would appear as a bright dust ring. +However, these findings may be altered by the dust diffusion due +10-2 +10-1 +100 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +10-2 +10-1 +100 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +10-2 +10-1 +100 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +Figure 7. Density profile of dust particles of different sizes ranging from +100 𝜇m to 1 cm with the planets in a 2:1 resonance. These profiles are +compared to the re-scaled gas density (multiplied by 0.0033). The top panel +is for a kinematic viscosity equal to 𝜈 = 10−6 𝑟2 +0 Ω0 and the middle panel for +𝜈 = 10−7 𝑟2 +0 Ω0. The bottom panel refers to a model with 𝜈 = 10−6 𝑟2 +0 Ω0, but +without diffusion and back-reaction. +MNRAS 000, 1–9 (2023) + +8 +Marzari and D’Angelo +Figure 8. Density maps illustrating gas and dust distributions around planets +locked in the 2:1 orbital resonance (𝜈 = 10−7 𝑟2 +0 Ω0) . The top panel refers +to the surface density of the gas. The other panels refer, respectively, to the +distributions of 100 𝜇m, 1 mm and 1 cm dust particles. +10-2 +10-1 +100 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 +Σ (g/cm2) +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +10-2 +10-1 +100 + 2 + 4 + 6 + 8 + 10 + 12 +Σ g/cm2 +r (au) +s=100 um +s=1 mm +s=1 cm +gas-sc +planets +Figure 9. Dust and gas density profiles (as in Figure 7) around a pair of +planets locked in a 2:1 orbital resonance, after 4 × 104 yrs, in an 𝛼-viscosity +disc. In the top panel 𝛼 = 10−3 and in the bottom panel 𝛼 = 10−4. +to gas turbulence and by a change in gap morphology due to the +back-reaction of the dust on the gas. It is, therefore, important to test +if these two processes weaken the dust trap at the outer edge of the +gap, allowing the dust to filter through the gap and preventing the +formation of an inner dust cavity and of the outer over-density ring. +We used the code FARGO3D (Benítez-Llambay et al. 2019), in +which dust species are treated as additional pressureless fluids, and +performed a set of local-isothermal, high resolution simulations in- +volving giant planets locked in the 3:2 and 2:1 orbital resonances +and including the effects of diffusion and dust back-reaction. For the +3:2 resonance, the diffusion and back-reaction slightly affect the dust +distribution by reducing the height of the overdense region at the +outer edge of the gas gap but without removing it. This has been +tested for different values of the disc viscosity, which determines the +amount of diffusion. This outcome proves that the dust trap is still +strong enough to halt the inward drift of the dust and to lead to the +formation of an inner cavity in the dust distribution and an overdense +ring at the outer edge of the gap. The details of peak heights can also +be affected by continued supply of solids from large radial distances +and, therefore, depend on boundary effects and evolution timescales. +Given the limited radial extent of the models presented herein, the +possible feedback of very dense rings of solids on the gas distribution +was not investigated. +MNRAS 000, 1–9 (2023) + +9 +102 +5 +4 +azim (rad) +(z g/6) +3 +10 +M +2 +1 +10 0 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)9 +5 +10 0 +4 +azim (rad) +(g/cm 2) +3 +10-1 +M +2 +1 +0 +10-2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)9 +5 +10 0 +4 +azim (rad) +(g/cm 2) +3 +10-1 +M +2 +L +0 +10-2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)9 +10 0 +5 +4 +azim (rad) +3 +10-1 +2 +1 +0 +10-2 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +r(au)Diffusion and back-reaction effects on dust +9 +The 2:1 resonance also provides a robust mechanism capable of +creating an efficient dust trap. The height of the outer peak (outside +the planets’ orbits) is not much affected by diffusion. Its morphology, +however, appears more complex than it does in the 3:2 resonance +situation, since in some cases two dust peaks form at the outer edge +of the gas gap, possibly due to the development of multiple dust +traps. The robustness of these results for this resonance were tested +by performing two additional simulations in which we switched from +a constant kinematic viscosity, 𝜈, to a constant 𝛼 viscosity parameter. +In terms of large-scale features, the outcome of these simulations do +not significantly differ from those with constant 𝜈, showing that the +formation of the inner dust cavity and the outer peak(s) are not due +to the viscosity parametrization (although details can depend on the +type of viscosity). +In the models presented herein, the gas distribution interior to +the planets’ orbits is not significantly depleted. However, ongoing +accretion on the planets, neglected here, is expected to reduce the gas +mass flux toward the inner disc (Lubow & D’Angelo 2006), possibly +leading to the formation of an inner cavity in the gas distribution as +well. +ACKNOWLEDGEMENTS +We thank the reviewer, Clément Baruteau, whose comments helped +us improve this paper. GD acknowledges support provided by NASA’s +Research Opportunities in Space and Earth Science. Computational +resources supporting this work were provided by the NASA High- +End Computing (HEC) Program through the NASA Advanced Su- +percomputing (NAS) Division at Ames Research Center. +DATA AVAILABILITY +The data underlying the research results described in the article will +be shared upon reasonable request to the authors. +REFERENCES +Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214 +Balbus S. A., Hawley J. F., 1998, Reviews of Modern Physics, 70, 1 +Balbus S. A., Hawley J. 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R., Kley W., Picogna G., 2017, A&A, 599, L6 +Taki T., Fujimoto M., Ida S., 2016, A&A, 591, A86 +Turrini D., Marzari F., Polychroni D., Testi L., 2019, ApJ, 877, 50 +Urpin V., 2003, A&A, 404, 397 +Woitke P., Helling C., 2003, A&A, 399, 297 +MNRAS 000, 1–9 (2023) + diff --git a/OdFRT4oBgHgl3EQfHjfc/content/tmp_files/load_file.txt b/OdFRT4oBgHgl3EQfHjfc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b0d27c1feec6f145ffe1364ea72fa460af1c023 --- /dev/null +++ b/OdFRT4oBgHgl3EQfHjfc/content/tmp_files/load_file.txt @@ -0,0 +1,658 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf,len=657 +page_content='MNRAS 000, 1–9 (2023) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='0 Effects of turbulent diffusion and back-reaction on the dust distribution around two resonant planets Francesco Marzari1★ and Gennaro D’Angelo,2† 1Department of Physics and Astronomy, University of Padova, via Marzolo 8, I-35131, Padova, Italy 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Accepted 1 February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Received 1 February 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' in original form 1 February 2023 ABSTRACT In evolved and dusty circumstellar discs, two planets with masses comparable to Jupiter and Saturn that migrate outwards while maintaining an orbital resonance can produce distinctive features in the dust distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Dust accumulates at the outer edge of the common gas gap, which behaves as a dust trap, where the local dust concentration is significantly enhanced by the planets’ outward motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Concurrently, an expanding cavity forms in the dust distribution inside the planets’ orbits, because dust does not filter through the common gaseous gap and grain depletion in the region continues via inward drifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' There is no cavity in the gas distribution because gas can filter through the gap, although ongoing gas accretion on the planets can reduce the gas density in the inner disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Such behaviour was demonstrated by means of simulations neglecting the effects of dust diffusion due to turbulence and of dust backreaction on the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Both effects may alter the formation of the dust peak at the gap outer edge and of the inner dust cavity, by letting grains filter through the dust trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' We performed high resolution hydrodynamical simulations of the coupled evolution of gas and dust species, the latter treated as pressureless fluids, in the presence of two giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' We show that diffusion and backreaction can change some morphological aspects of the dust distribution but do not alter some main features, such as the outer peak and the expanding inner cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' These findings are confirmed for different parametrizations of gas viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Key words: accretion, accretion discs — methods: numerical — planets and satellites: gaseous planets — planet–disc interactions 1 INTRODUCTION In a recent study, Marzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2019) examined the distribution of dust particles around two resonant planets embedded in a circum- stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The two planets, with masses comparable to those of Jupiter and Saturn, had orbits in a resonant configuration, with ratios of the mean motion equal to either 2:1 or 3:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Because of the type of resonance and of the applied disc conditions, the planets tend to migrate outwards and dust particles tend to accumulate outside of the orbit of the exterior planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Concurrently, the inward migration of dust grains that move inside of the orbit of the interior planet leads to an enlargement of the dust gap compared to the gap in the gas and to a dynamical decoupling between the gaps in the gas and dust distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The build-up of the dust density at the outer edge of the gap surrounding the planets is markedly higher in the case of the 2:1 mean-motion resonance and may appear as a bright ring (at appropriate wavelengths) in resolved observations of discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' A similar phenomenon was also found for lower-mass planets (in the Super-Earth mass range, Marzari & D’Angelo 2020), although less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' All those simulations were performed without includ- ing the effects of possible dust diffusion due to gas turbulence and of dust back-reaction on the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Here we explore the consequences ★ E-mail: francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='marzari@pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='it † E-mail: gennaro@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='gov of these two mechanism on the accumulation of dust grains at the outer edge of the gap and on the formation of a wider gap in the dust distribution compared to the density gap in the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This latter feature may lead to the formation of a transition disc, if the planets are close enough to the star (low gas surface density inside the planets’ orbits can be caused by ongoing accretion of gas on star and planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Circumstellar discs are likely turbulent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Various mechanisms have been proposed as drivers of turbulence, such as convective instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', Klahr & Hubbard 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Lyra 2014), vertical shear instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', Urpin 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Stoll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2017), and magneto-rotational instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', Balbus & Hawley 1991, 1998, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Gas turbulence may force dust grains to diffuse (over a length-scale dictated by the type of turbulence), a pro- cess that not only may affect dust accumulation but can also hinder the efficiency of dust entrapment at radial location of gas pressure maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In fact, according to Sierra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2019) and Pinilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2020), dust diffusion may reduce, or even prevent, significant con- centrations of grains at locations of gas “bumps”, since the grains can disperse out of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This process might affect the conclusions of our previous results on the accumulation of dust at the outer edge of the gaseous gap of two planets in resonance by letting dust seep through the gap and reach the inner disc regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' If the effect is large enough, the concentration of dust at those radial locations, obtained in the numerical simulations by Marzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2019) and Marzari & D’Angelo (2020), may be severely depleted or even largely absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='13489v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='EP] 31 Jan 2023 2 Marzari and D’Angelo In addition to dust diffusion, the back-reaction of dust on gas can also impact the formation of grain concentration at a local pressure maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' According to Taki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2016), dust back-reaction can deform the pressure gradient of the gas when high-enough values of the dust-to-gas mass ratio are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This may be the case of the dust concentration attained at the outer edge of gaseous gaps, observed in the simulations of two planets in resonance migrating away from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' To test the relevance of these two mechanisms, diffusion and back- reaction, on the formation of dust over-dense regions caused by the outward migration of two planets in resonance, we performed simula- tions of the evolution of two planets in resonance in which both these two mechanisms are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In Section 2, we describe the numerical model exploited to study the coupled evolution of dust and gas in presence of the two resonant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In Section 3, we outline the dust behaviour when the planets evolve in the 3:2 mean- motion resonance whereas, in Section 4, we analyse the case of the 2:1 mean-motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In Section 5, we test the robustness of our results by changing the viscosity parameterization, including a constant kinematic viscosity the one that applies a constant value of the 𝛼 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Finally, in Section 6, we discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2 METHODS AND ALGORITHMS In past work on the coupled evolution of dust and gas in proto- planetary discs, we adopted a Lagrangian description of the solid component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Instead, an Eulerian formalism is applied in the present study, since solids are treated as pressureless fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Some details on the involved equations are provided below to highlight the differ- ences between the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Marzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2019) and Marzari & D’Angelo (2020) used the two-dimensional (2D) FARGO hydro- dynamics code (Masset 2000), modified to include the dynamical evolution of dust particles embedded in the gaseous disc in a La- grangian fashion (Picogna & Kley 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The drag force on the particles was computed from the local gas density according to the equation (Woitke & Helling 2003) F = � 3𝐾 3𝐾 + 1 �2 FE + � 1 3𝐾 + 1 �2 FS, (1) where FE is the Epstein drag contribution, given by FE = −4 3 � 1 + 9𝜋 128 𝑀2 �1/2 𝑠2𝜌𝑔𝑣thvrel, (2) and FS is the Stokes drag component, given by FS = −1 2𝐶𝐷𝜋𝑠2𝜌𝑔𝑣relvrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (3) In the above equations, 𝜌𝑔 is the gas density, 𝑠 is the radius of the particle, 𝑣th is the local thermal velocity of the gas and vrel is the velocity of the dust particle relative to the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The quantity 𝐾 is the Knudsen number and 𝑀 is the Mach number (computed from the particle’s relative velocity 𝑣rel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Quantity 𝐶𝐷 is the drag coefficient for the Stokes regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In this paper, to test the relevance of diffusion and back-reaction on the formation of dust over-dense regions caused by the outward migration of two planets in resonance, we carry out simulations with the code FARGO3D (Benítez-Llambay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In this version of the code, the dust particles are treated as additional pressureless fluids where momentum is transferred between the gas and each of the dust species (but not among dust species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The dust fluid is affected by Epstein drag, which imparts a force per unit volume to a dust species given by F𝑑 = −𝜌𝑑 Ω 𝜏𝑠 (v𝑑 − v𝑔), (4) where 𝜌𝑑 is the dust density, 𝜏𝑠 is the Stokes number of the particle and Ω is the Keplerian frequency of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' An equal and opposite force is imparted to the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Note that, in Equation (4), information regarding the drag coefficient are incorporated into 𝜏𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' A term is added to the continuity equation to model the diffusion of the dust species within the gas (Morfill & Voelk 1984) 𝜕𝜌𝑑 𝜕𝑡 = ∇ · � 𝐷𝑑𝜌∇ 𝜌𝑑 𝜌 � , (5) where 𝜌 = 𝜌𝑔 + 𝜌𝑑 and 𝜌𝑑 is the density of individual dust species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Equation (5) is only applied to the pressureless fluids and it assumes the same diffusion coefficient for all dust species, which is taken equal to the value of the gas kinematic viscosity (Brauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2008) 𝐷𝑑 = 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (6) The effects of this choice are not tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The original version of the code applies Stokes numbers, 𝜏𝑠, in Equation (4) that are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The code was modified so that we can select each pressureless fluid (dust species) not according to a Stokes number, which varies as a function of the local properties of the gas (density, temperature and velocity), but rather according to the particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 3 DUST DISTRIBUTION NEAR PLANETS IN THE 3:2 RESONANCE We first investigate the case of a pair of planets that become captured in the 3:2 mean-motion resonance and migrate outward thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The interior planet has the mass of Jupiter whereas the exterior planet has the mass of Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' A more massive inner planet is a condition conducive to outward migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The planets orbit in a cold, local-isothermal disc with a fixed aspect ratio 𝐻/𝑟 = ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (Calculations with a larger ratio, ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='05, are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=') The disc extends from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='4 to 12 au and is discretised over a grid of 512 × 1024 area elements (in the radius and azimuth, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The initial surface density of the gas declines as Σ(𝑟) = Σ0 �𝑟0 𝑟 � , (7) where Σ0 = 200 g cm−2 is the density at the reference radius 𝑟0 = 1 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Three different populations of icy grains (bulk density 1 g cm−3) are included in the simulations, whose sizes are 100 𝜇m, 1 mm and 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For the applied disc conditions, these particles have Stokes numbers less than ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The initial dust-to-gas mass ratio for each of the three dust species is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='0033, so that the overall dust-to-gas mass ratio adds up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='01, which is a typical value adopted for circumstellar discs and is based on values found in the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' However, dust needs not be primordial in origin, that is, part of the inventory of solids from which the planets formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In fact, the dust populations may represent, or contain, second generation dust produced by collisions among left- over planetesimals, after the planets became massive enough (Turrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' D’Angelo & Marzari 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Marta Bernabò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The equations describing the evolution of the system are solved in a non-inertial reference frame centered on the star, including the indirect terms arising from the planets’ and disc’s gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' MNRAS 000, 1–9 (2023) Diffusion and back-reaction effects on dust 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='00 0 20 40 60 80 100 120 a (au) Time (Kyr) ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-6 ν =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-6 ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='16 0 20 40 60 80 100 120 Eccentricity Time (Kyr) ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-6 ν =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-6 ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The top panel shows the semi-major axis of pairs of planets during their migration, for three values of a constant kinematic viscosity, 𝜈, of the gas (in units of 𝑟2 0 Ω0, see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In these cases, the planet pair is locked in the 3:2 mean-motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the bottom panel illustrates the evolution of the orbital eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The inner, more massive planet has a lower eccentricity in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Data are averaged over a 250 yr window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' see text for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Various values of the gas kinematic viscosity, 𝜈, are considered be- cause this parameter affects the tidal interactions between the planets and the gas, and also determines the amount of dust diffusion through Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Additionally, it can also affect the speed of the planets’ outward migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In these models, we adopt a constant value of kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The impact of 𝜈 on the efficiency of the outward migration is illustrated in Figure 1, for a given value of the initial gas density at the reference radius 𝑟0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In these models, the planets start to migrate at the beginning of the simulations, when the distributions of gas and dust are unperturbed (hence the initial rapid inward migra- tion of the planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' As the outer planet approaches the inner planet and their orbits become caught in resonance, the tidal perturbations exerted by the outer planet on the circumstellar material alter the torque balance on the inner planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Consequently, the inner planet first slows down and then migrates away from the star, pushing the outer planet outward through the resonance forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The evolution of the semi-major axis of the outer planet, 𝑎2, is shown for three different values of the gas kinematic viscosity: 𝜈 = 10−6, 5 × 10−6, and 10−5, in units of 𝑟2 0Ω0 (Ω0 is the Ke- plerian frequency at 𝑟0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Note that a constant kinematic viscosity corresponds to a variable 𝛼 parameter, 𝛼 ∝ 𝜈/(ℎ2√𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' With our choice of parameters, at 𝑟 = 𝑟0, said values of 𝜈 would correspond to 𝛼0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='0025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='0125, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='025, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' However, in the disc regions where the planets orbit, 𝛼 would be smaller by a factor of 2 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' After a different behaviour at the beginning of the evolution, prior to or shortly after the capture into the 3:2 orbital resonance (see Figure 1), the planets undergo sustained outward migration locked in mean-motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The migration speed of the pair is related to 𝜈 and determined by the shape of the common (or overlapping) gaseous gap of the two planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' A similar outcome is obtained for the cases involving the 2:1 resonance, as shown in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Notice that the outer planet is subject to a negative torque exerted by the disc material exterior to its orbit, hence it would tend to move inward, whereas the resonance forcing pushes it outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' These two opposing torques allow the resonance to be maintained during the outward migration of the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the calculations presented herein, all disc material exerts torques on the planets, including material within the planets’ Hill spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Since the numerical resolution is limited and density variations close to the planets may not be properly resolved, some spurious effects may arise that affect outward migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' To quantify possible dif- ferences, one model was also performed by removing the torques exerted from within the planets’ Hill spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The orbital evolution is comparable to that of the calculation with default setup, although at a somewhat reduced outward migration speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The amount of kine- matic viscosity can also alter the local distribution of material around the planets and unresolved density gradients can possibly impact the resulting migration velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Nonetheless, it must be pointed out that for the purposes of this study the details of the outward migration process are not important, henceforth tests on the response of the system to numerical parameters are unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The only require- ment is that the planet pair becomes locked in resonance and moves away from the star for a prolonged amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' When the outer planet reaches 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='5 au from the star, we compare the dust distributions in the three cases with different viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This comparison is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The dust density profiles show a peak at the outer border of the gas gap carved by the planets’ tidal perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This peak is more marked in the distributions of the largest grains, which are less coupled to the gas (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', they have a larger Stokes number and therefore a longer coupling timescale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Dust-to-gas mass ratios at the peak location range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='04 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='08, increasing as 𝜈 decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the region inside the inner edge of the gap there is a significant depletion of dust due to drifting motion towards the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Re-supply of dust to this region is reduced, or halted, by the dust trap at the outer edge of the gas gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' After some time, the disc would appear as a transition disc with an inner cavity in the dust density, which expands outward over time due to the outward migration of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' These effects appear more evident at lower gas viscosity, which may be due to the lower level of diffusion but it may also be related to the different migration velocity of the planet pair (see Figure 1), as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Beyond ≈ 10 au the disc is depleted of mm- and cm-grains (but not of the smallest grains), an effect associated to the inward drift of the particles via gas drag (which does not affect as much the smallest grains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This is a boundary effect related to the fact that particles are not flowing inward from greater distances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', there is not re-supply of solids at the outer boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Test simulations not reported here confirm this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The accumulation efficiency of particles at the outer edge of the gas gap depends on the stopping time of the particles (𝜏𝑠/Ω) and the timescale over which the radial pressure gradient of the gas moves (in our case, due to planet migration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For a given stopping time, the shorter the outward migration timescale, the less time dust grains have to accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Stated differently, for a similar orbital MNRAS 000, 1–9 (2023) 4 Marzari and D’Angelo 10-2 10-1 100 2 4 6 8 10 12 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets 10-2 10-1 100 2 4 6 8 10 12 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets 10-2 10-1 100 2 4 6 8 10 12 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Surface density (averaged in azimuth around the star) of dust particles of different sizes, ranging from 100 𝜇m to 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' These profiles are compared to the re-scaled gas density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='0033;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' the total dust-to-gas mass ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The top panel refers to a kinematic viscosity equal to 𝜈 = 10−6, the middle panel to 𝜈 = 5 × 10−6 and the bottom panel to 𝜈 = 10−5, in units of 𝑟2 0 Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 10-2 10-1 100 2 4 6 8 10 12 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Dust and gas profiles (as in Figure 2) with 𝜈 = 10−5 𝑟2 0 Ω0, but without the inclusion of diffusion and back-reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' configuration of the planets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', similar orbital frequencies Ω, a more rapid outward migration can facilitate the filtering process of dust toward the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The overall outcome is a less depleted inner disc in the cases with more vigorous outward migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For the same reason, the different migration speed also controls the build up of the dust at the outer edge of the gas gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' As a consequence, the reduction in the peak density at the outer edge of the gap observed at higher viscosity values, shown in Figure 2, is likely due to the combination of the two effects: a higher diffusion rate and a decrease in the trapping efficiency of dust grains caused by the faster outward migration rate of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' To better characterize the role of diffusion and back-reaction in the formation, size and shape of the outer peak in dust density, we ran a simulation in which both these two processes were neglected, adopting a kinematic viscosity 𝜈 = 10−5 𝑟2 0Ω0 (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In this model, the migration speed is broadly consistent with that of the model illustrated in the bottom panel of Figure 2, but the peak in the dust density distributions of the largest particles tend to be higher and sharper compared to corresponding one in the the bottom panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The implication is that diffusion is indeed affecting the concentration of the dust by spreading large grains over a broader radial region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Nonetheless, the reduced efficiency in collecting dust caused by diffusion is not strong enough to prevent the formation of a prominent density peak outside the planets’ orbits, which may be detected by high resolution observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In Figure 4 we plot the surface density distribution of the gas and of dust grains of different sizes for the case with lowest viscosity, 𝜈 = 10−6 𝑟2 0Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The gas gap is significantly narrower than that of the dust and the width of the latter is larger for larger grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For 1 cm size dust grains, the density distribution is confined in an over-dense ring at the outer edge of the gas gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Therefore, according to these results, dust diffusion and back-reaction are not able to prevent the formation of narrow rings in the dust distribution at the outer edge of the gas gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In our simulations there is no re-supply of dust at the outer boundary of the simulated disc, but it is expected that a continuous distribution of solids beyond the grid boundary would supply dust to the inner disc regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In this case, at the outer edge of the gap, we would observe an enhanced density, as predicted by our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Beyond this density peak, however, there would be a continuous distribution of dust originating from more distant regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Over timescales much longer than those simulated here, dust drifting MNRAS 000, 1–9 (2023) Diffusion and back-reaction effects on dust 5 from larger distances may also affect the density peaks at the outer edge of the gas gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In fact, an additional simulation with a wider radial boundary (not reported here) does show enhanced peaks in large grains, due to solids drifting from farther distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Also the population of small grains would increase at the peaks over longer times, but at a slower rate, dictated by the drift velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Nonetheless, the depletion of dust within the inner edge of the gas gap would not be affected by this process because the outer dust trap appears to be efficient enough to halt (or severely impede) refilling of grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In fact, if refilling of the inner disc was sustained, it would occur within the time of our simulations but it is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Continued supply of dust from farther out in the disc may, at some point, raise the density in the peak regions beyond some threshold value to make the peaks unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For example, the dust-to-gas mass ratio may become large enough to induce a back-reaction response on the gas that redistributes the particles over some radial region via collisions and/or enhanced dust coagulation may ensue (coagulation into larger particles would reduce the back-reaction force exerted on the gas, because of the lower surface-to-mass ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The back- reaction response may also smooth out the gas density gradient in the radial direction, altering the radial pressure gradient and reducing its ability to retain grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Such processes are not considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' To test the impact of a larger aspect ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=', warmer disc) on the formation of the inner dust cavity and of the peak external to the gas gap, we performed two additional simulations adopting ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the first model we used a higher gas surface density (Σ0 = 800 g cm−2), in order to increase the speed of planet, migration while in the second we used the same density as in the previous cases (Σ0 = 200 g cm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the latter simulation, since the rate of outward migration is smaller, the planets are located closer to the star in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the top panel of Figure 5, the density profiles are shown for the different grain sizes in the high density case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The bottom panel illustrates those profiles for the low density case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In both simulations the density patterns are similar to those in Figure 2, suggesting that a higher aspect ratio does not impair the ability of two planets in resonance to carve an inner gap in the distribution of the larger grains and to trap dust at the gap’s outer edge (dust-to-gas mass ratios around the peak region are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Thus, dust filtering through the outer edge of the gas gap is not increased by the different morphology of the dust trap in these warmer discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 4 DUST DISTRIBUTION NEAR PLANETS IN THE 2:1 RESONANCE To test the evolution of the dust when the planets are captured in a 2:1 mean-motion resonance, we decreased the gas density to Σ0 = 40 g/cm2 in order to induce orbital locking in this resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' It is known that capture in this mean-motion resonance is a more delicate process than capture in the 3:2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' If the relative migration velocity with which the pair of planets approach each other is above a certain threshold, the resonance forcing is overcome and convergent migration continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Additionally, once the 2:1 orbital resonance is established, migration of the two planets typically proceeds inward because of the way the two gas gaps overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In this configuration, outward migration may be obtained by choosing an appropriately low kinematic viscosity so that a wide, common gaseous gap forms around the orbits of the two planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' As in the the models of the previous section, the planets start to migrate in unperturbed distributions of gas and therefore the initial inward migration of the planets is artificially rapid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This choice af- Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Surface density maps illustrating gas and dust distributions around two planets locked in the 3:2 resonance (𝜈 = 10−6 𝑟2 0 Ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The top panel shows the gas density, whereas the second, third and fourth panels represent 100 𝜇m, 1 mm and 1 cm particles, respectively.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='r(au)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='Marzari and D’Angelo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='Σ (g/cm2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='r (au) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='s=100 um ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='s=1 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='s=1 cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='gas-sc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='planets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='Σ (g/cm2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='r (au) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='s=100 um ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='s=1 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='s=1 cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='gas-sc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='planets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Dust and gas density profiles (as in Figure 2) in a disc with 𝜈 = 10−5 𝑟2 0 Ω0 and aspect ratio 𝐻/𝑟 = ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the top panel Σ0 = 800 g cm−2 whereas Σ0 = 200 g cm−2 in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' fects the radius at which the planets are trapped in resonance but is otherwise not much relevant for the purposes of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' As for the models discussed in Section 3, the model with a larger value of the kinematic viscosity results in a more vigorous outward migra- tion of the planets once they become trapped in resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This is illustrated in Figure 6 for cases with 𝜈 = 10−6 and 𝜈 = 10−7 𝑟2 0Ω0 (𝛼0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='5 × 10−3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='5 × 10−4, respectively) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Note, however, that the orbits also become significantly eccentric, which also affects out- ward migration (D’Angelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' We also tested a larger value, 𝜈 = 5 × 10−6 𝑟2 0Ω0, but the outer planet crosses the 2:1 resonance with the inner planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The pair is temporarily trapped in the 5:3 res- onance at which point it begins migrating outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' That resonance is then broken and the pair becomes finally captured in the 3:2 reso- nance, continuing to migrate outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For even higher values of the kinematic viscosity, there is no trapping in the 2:1 resonance, which might be attained by further reducing Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' However, this possibility was not tested since it could lead to initial dust-to-gas mass ratios too dissimilar from the other simulations presented herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The dust distributions are shown in Figure 7 for the two different values of kinematic viscosity and, in the bottom panel, for a model without the inclusion of diffusion and back-reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Because of the significantly different migration speed, we only compare the dust density distributions at the end of the simulation (when the planet are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='50 0 10 20 30 40 50 60 70 a (au) Time (Kyr) ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-6 ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='15 0 10 20 30 40 50 60 70 Eccentricity Time (Kyr) ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-6 ν =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='e-7 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The top panel shows the migration of the exterior planet of a pair, for two values of a constant kinematic viscosity of the gas, 𝜈, as indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The planet pair is locked in the 2:1 mean-motion resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Only the semi-major axis of the exterior planet is plotted because of the larger mutual distance compared, for example, to the 3:2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The orbital eccentricity of both interior and exterior planets is illustrated in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Data are averaged over a 250 yr window as in the case of the 3:2 resonance (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' located at different orbital radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For this resonance too a significant dust enhancement develops at the outer edge of the gas gap, where dust-to-gas mass ratios can achieve values of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The inner cavity is larger at all grain sizes, compared to that produced by the 3:2 resonant configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This is possibly related to the slower outward migration rate of the planets in the 2:1 resonance, which can reduce filtering of the dust through the orbits of the planets and toward the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Additionally, the lower gas density increases the Stokes numbers of the grains, reducing their drift timescale and facilitating grain removal from the inner disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Comparing the top and bottom panels of Figure 7, one can notice significant differences in the dusty features exterior to the planets orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Since the gas density is also different in the two models (at those times), it is unclear how much of the difference is caused by the action of diffusion and back-reaction of the solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The orbital eccentricity of the planets can drive asymmetries in the distribution of the disc material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Consequently, both the gas gap outer edge and dusty rings can become asymmetric (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' We did not perform a detailed analysis of ring asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' However, one possible explanation for the more asymmetric features arising from the planets in the 2:1 resonance (compared to those in the 3:2 resonance, see Figure 4), may be the larger eccentric perturbation MNRAS 000, 1–9 (2023) Diffusion and back-reaction effects on dust 7 driven by the inner (more massive) planet, which tends to have a larger orbital eccentricity in the 2:1 resonance than in the 3:2 resonance (the outer planets can have comparable eccentricities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In Figure 8, the dust density distributions are shown for grains of various sizes for the case with lowest viscosity, 𝜈 = 10−7 𝑟2 0Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' As mentioned, the width of the gap is significantly larger compared to the case of the 3:2 resonance and the dusty ring at the outer edge of the gas gap is evident for all particles, more than in the 3:2 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For the 2:1 resonance, the effects of the dust trap appear more marked, both at the inner and at the outer edge of the gas gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' It is expected that after sufficient time from the beginning of the outward migration, the disc reduces to a single overdense ring at the outer edge of the gap, close to the outer planet orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' As discussed in the previous section, this feature would result from the lack of dust supply from larger orbital radii, beyond the outer boundary of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 5 COMPARISON WITH ALPHA VISCOSITY To test the robustness of our results, we performed two additional simulations, for the 2:1 resonance, in which the kinematic viscosity is set as 𝜈 = 𝛼𝐻2Ω, where 𝐻 is the local pressure scale height of the disc and the parameter 𝛼 is assumed to be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Since we work with a local-isothermal disc and a constant aspect ratio, 𝐻 ∝ 𝑟 and 𝜈 ∝ 𝛼√𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the first model, we set 𝛼 = 10−3 whereas we set 𝛼 = 10−4 in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' These parameters result in 𝜈 ≈ 10−6 and ≈ 10−7 𝑟2 0Ω0 around the middle radius of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The outcomes of these simulations are illustrated in Figure 9, after 4 × 104 yrs of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The density peak at the outer edge of the gas gap is clearly visible in both cases, although there are differences in over-density morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the top panel, at 𝛼 = 10−3, the peak appears similar at all sizes while, in the bottom panel (𝛼 = 10−4), the peak is split in two for the largest particles (𝑠 = 1 mm and 𝑠 = 1 cm) and composed of three separate maxima for 𝑠 = 100 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This behaviour was not observed for the 3:2 resonance but it was already present in the simulation involving the 2:1 mean- motion resonance obtained with a constant kinematic viscosity (see Figure 7, top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' A possible interpretation is that for the 2:1 resonance multiple dust traps develop at the outer border of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' By comparing the gas density distribution in the two cases shown in Figure 9, this hypothesis appears to be confirmed since, in the case with 𝛼 = 10−4, Σ at the outer border of the gas gap appears more variable compared to that of the case with 𝛼 = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The splitting of the peak in various maxima can be also observed in the simulations of Marzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2019), for the case involving a pair of planet locked in the 2:1 resonance, even if their physical model (radiative disc), the initial parameters for the gas density and the viscosity values are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 6 CONCLUSIONS According to the models presented by Marzari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' (2019), two planets in the mass range of Jupiter and Saturn, embedded in a circumstellar disc and migratingoutwards,canstronglyaffect the dust distribution in their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' They can create a transition disc with an inner cavity in the dust distribution that expands following the outward migration of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' They can also build a strong peak in the dust density at the outer edge of the gas gap carved by the resonant planets, which would appear as a bright dust ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' However, these findings may be altered by the dust diffusion due 10-2 10-1 100 2 3 4 5 6 7 8 9 10 11 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets 10-2 10-1 100 2 3 4 5 6 7 8 9 10 11 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets 10-2 10-1 100 2 3 4 5 6 7 8 9 10 11 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Density profile of dust particles of different sizes ranging from 100 𝜇m to 1 cm with the planets in a 2:1 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' These profiles are compared to the re-scaled gas density (multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='0033).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The top panel is for a kinematic viscosity equal to 𝜈 = 10−6 𝑟2 0 Ω0 and the middle panel for 𝜈 = 10−7 𝑟2 0 Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The bottom panel refers to a model with 𝜈 = 10−6 𝑟2 0 Ω0, but without diffusion and back-reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' MNRAS 000, 1–9 (2023) 8 Marzari and D’Angelo Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Density maps illustrating gas and dust distributions around planets locked in the 2:1 orbital resonance (𝜈 = 10−7 𝑟2 0 Ω0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The top panel refers to the surface density of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The other panels refer, respectively, to the distributions of 100 𝜇m, 1 mm and 1 cm dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 10-2 10-1 100 2 3 4 5 6 7 8 9 10 11 Σ (g/cm2) r (au) s=100 um s=1 mm s=1 cm gas-sc planets 10-2 10-1 100 2 4 6 8 10 12 Σ g/cm2 r (au) s=100 um s=1 mm s=1 cm gas-sc planets Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Dust and gas density profiles (as in Figure 7) around a pair of planets locked in a 2:1 orbital resonance, after 4 × 104 yrs, in an 𝛼-viscosity disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the top panel 𝛼 = 10−3 and in the bottom panel 𝛼 = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' to gas turbulence and by a change in gap morphology due to the back-reaction of the dust on the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' It is, therefore, important to test if these two processes weaken the dust trap at the outer edge of the gap, allowing the dust to filter through the gap and preventing the formation of an inner dust cavity and of the outer over-density ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' We used the code FARGO3D (Benítez-Llambay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 2019), in which dust species are treated as additional pressureless fluids, and performed a set of local-isothermal, high resolution simulations in- volving giant planets locked in the 3:2 and 2:1 orbital resonances and including the effects of diffusion and dust back-reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' For the 3:2 resonance, the diffusion and back-reaction slightly affect the dust distribution by reducing the height of the overdense region at the outer edge of the gas gap but without removing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This has been tested for different values of the disc viscosity, which determines the amount of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' This outcome proves that the dust trap is still strong enough to halt the inward drift of the dust and to lead to the formation of an inner cavity in the dust distribution and an overdense ring at the outer edge of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The details of peak heights can also be affected by continued supply of solids from large radial distances and, therefore, depend on boundary effects and evolution timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Given the limited radial extent of the models presented herein, the possible feedback of very dense rings of solids on the gas distribution was not investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' 1–9 (2023) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='r(au)Diffusion and back-reaction effects on dust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='The 2:1 resonance also provides a robust mechanism capable of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content='creating an efficient dust trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The height of the outer peak (outside the planets’ orbits) is not much affected by diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Its morphology, however, appears more complex than it does in the 3:2 resonance situation, since in some cases two dust peaks form at the outer edge of the gas gap, possibly due to the development of multiple dust traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' The robustness of these results for this resonance were tested by performing two additional simulations in which we switched from a constant kinematic viscosity, 𝜈, to a constant 𝛼 viscosity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In terms of large-scale features, the outcome of these simulations do not significantly differ from those with constant 𝜈, showing that the formation of the inner dust cavity and the outer peak(s) are not due to the viscosity parametrization (although details can depend on the type of viscosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' In the models presented herein, the gas distribution interior to the planets’ orbits is not significantly depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' However, ongoing accretion on the planets, neglected here, is expected to reduce the gas mass flux toward the inner disc (Lubow & D’Angelo 2006), possibly leading to the formation of an inner cavity in the gas distribution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the reviewer, Clément Baruteau, whose comments helped us improve this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' GD acknowledges support provided by NASA’s Research Opportunities in Space and Earth Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQfHjfc/content/2301.13489v1.pdf'} +page_content=' Computational resources supporting this work were provided by the NASA High- End Computing (HEC) Program through the NASA Advanced Su- percomputing (NAS) Division at Ames Research Center.' metadata={'source': 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b/StAyT4oBgHgl3EQf8PpF/content/tmp_files/2301.00852v1.pdf.txt @@ -0,0 +1,1122 @@ +Draft version January 4, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Proton and electron temperatures in the solar wind and their correlations with the solar wind speed +Chen Shi (时辰) +,1 Marco Velli +,1 Roberto Lionello +,2 Nikos Sioulas +,1 Zesen Huang (黄泽森) +,1 +Jasper S. Halekas +,3 Anna Tenerani +,4 Victor R´eville +,5 Jean-Baptiste Dakeyo,6 Milan Maksimovi´c +,6 +and Stuart D. Bale +7, 8 +1Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles +Los Angeles, CA 90095, USA +2Predictive Science Inc +San Diego, CA 92121, USA +3Department of Physics and Astronomy, University of Iowa +Iowa City, IA 52242, USA +4Department of Physics, The University of Texas at Austin, +TX 78712, USA +5IRAP, Universit´e Toulouse III - Paul Sabatier, CNRS, CNES, Toulouse, France +6LESIA, Observatoire de Paris, Universit e PSL, CNRS, Sorbonne Universit e, Universit e de Paris, 5 place Jules Janssen, 92195 +Meudon, France +7Physics Department, University of California, Berkeley, CA 94720-7300, USA +8Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA +ABSTRACT +The heating and acceleration of the solar wind remains one of the fundamental unsolved problems in +heliophysics. It is usually observed that the proton temperature Ti is highly correlated with the solar +wind speed VSW , while the electron temperature Te shows anti-correlation or no clear correlation with +the solar wind speed. Here we inspect both Parker Solar Probe (PSP) and WIND data and compare +the observations with simulation results. PSP observations below 30 solar radii clearly show a positive +correlation between proton temperature and wind speed and a negative correlation between electron +temperature and wind speed. One year (2019) of WIND data confirm that proton temperature is +positively correlated with solar wind speed, but the electron temperature increases with the solar wind +speed for slow wind while it decreases with the solar wind speed for fast wind. Using a one-dimensional +Alfv´en-wave-driven solar wind model with different proton and electron temperatures, we for the first +time find that if most of the dissipated Alfv´en wave energy heats the ions instead of electrons, a positive +Ti − VSW correlation and a negative Te − VSW correlation arise naturally. If the electrons gain a small +but finite portion of the dissipated wave energy, the Te − VSW correlation evolves with radial distance +to the Sun such that the negative correlation gradually turns positive. The model results show that +Alfv´en waves are one of the possible explanations of the observed evolution of proton and electron +temperatures in the solar wind. +Keywords: Magnetohydrodynamics (1964), Solar wind (1534), Alfven waves (23) +1. INTRODUCTION +Solar wind is the plasma flow ejected from solar +corona, filling the interplanetary space. It carries a large +amount of mass and energy out of the Sun and serves +Corresponding author: Chen Shi +cshi1993@ucla.edu +as the medium of various physical processes and struc- +tures, such as waves and turbulence, coronal mass ejec- +tion, and magnetic reconnection. Solar wind continu- +ously interacts with the Earth and injects energy in the +Earth’s magnetosphere, causing strong disturbances of +the Earth’s magnetosphere. Understanding the gener- +ation of solar wind and the dynamics of the plasma in +the solar wind is necessary not only for a better space +arXiv:2301.00852v1 [astro-ph.SR] 2 Jan 2023 + +ID2 +Shi et al. +weather prediction but also for a deeper insight of the +fundamental plasma astrophysics. +A significant amount of energy is needed for the +plasma to escape the solar gravity and to accelerate +to a supersonic speed. +Early works (Parker 1958, +1964a,b, 1965) show that an isothermal solar corona +with strong thermal conduction can generate the ob- +served solar wind speeds. +However, in-situ measure- +ments imply that in the inner heliosphere both ion and +electron temperatures of the solar wind decay with ra- +dial distance to the Sun and the radial profiles of the +temperatures can be fitted with polytropic relations +(Marsch et al. 1982; Richardson et al. 1995; ˇStver´ak et al. +2015; Boldyrev et al. 2020; Shi et al. 2022). The poly- +tropic indices deduced from the in-situ measurements +are in general smaller than the adiabatic index 5/3, im- +plying that in-situ heating mechanisms are important. +One possible source of the in-situ heating is the tur- +bulence energy cascade. +In the solar wind, predomi- +nantly outward propagating Alfv´en waves exist (Belcher +& Davis Jr 1971), while smaller-amplitude inward prop- +agating Alfv´en waves are generated due to processes in- +cluding wave reflection induced by the gradient of Alfv´en +speed (Heinemann & Olbert 1980) and the large-scale +stream shears (Roberts et al. 1992; Shi et al. 2020). The +nonlinear interaction between the outward and inward +propagating waves leads to a turbulence energy cascade +(Kraichnan 1965), which eventually dissipates the en- +ergy of the electromagnetic fields into the plasma energy +through wave-particle interactions (Kasper et al. 2013; +Kobayashi et al. 2017) and intermittent structures (Os- +man et al. 2012; Matthaeus et al. 2015; Sioulas et al. +2022b). +It has been long observed that in the solar wind, +the proton temperature has a strong positive correla- +tion with the solar wind speed (Burlaga & Ogilvie 1973; +Lopez & Freeman 1986; Matthaeus et al. 2006; D´emoulin +2009; Elliott et al. 2012; Shi et al. 2021; Hofmeister et al. +2022). This positive correlation indicates the possibility +that a particular mechanism contributes to the proton +heating and solar wind momentum simultaneously. On +the contrary, electron core temperature usually shows +anti-correlation with the solar wind speed (Marsch et al. +1989; Halekas et al. 2020; Maksimovic et al. 2020) and +this anti-correlation may originate at the source of so- +lar wind due to interchange reconnection (Fisk 2003; +Gloeckler et al. 2003) as the coronal electron temper- +ature estimated from the fractions between heavy ions +measured in-situ, which is a good proxy for freeze-in +temperatures of the solar corona, has an anti-correlation +with solar wind speed (Geiss et al. 1995; Ko et al. 1997; +Gloeckler et al. 2003; von Steiger & Zurbuchen 2011). +Here, we propose a mechanism that the different +temperature-speed correlations for protons and elec- +trons are possibly generated in-situ as the solar wind +propagates. The major factor is how the energy source +for the acceleration of solar wind deposits differently +into protons and electrons. As the turbulence energy +cascades from MHD inertial scales toward ion kinetic +scales, various kinetic processes may arise and they +determine how the electromagnetic energy eventually +heats the ions and electrons. Two major processes are +kinetic Alfv´en waves (KAWs) which mainly heat the +electrons through Landau damping, and ion cyclotron +waves (ICWs) which heat the ions through cyclotron +resonance. Both KAWs (e.g. Podesta 2013; Salem et al. +2012) and ICWs (e.g. Jian et al. 2010, 2009) are identi- +fied in the solar wind, and recent observation made by +Parker Solar Probe shows that the two modes may coex- +ist (Huang et al. 2020). Gyrokinetic theory and simula- +tions show that the ratio between ion heating and elec- +tron heating during dissipation of Alfv´enic turbulence +is positively correlated with ion beta (ratio between ion +thermal pressure and magnetic pressure) (Howes et al. +2008; Schekochihin et al. 2009; Howes 2010; Kawazura +et al. 2019). Moreover, existence of compressive compo- +nent in the turbulence will increase the ratio between ion +and electron heating (Kawazura et al. 2020). Hybrid- +kinetic simulations using realistic solar wind parameters +at 1 AU show that 75-80% of the cascaded turbulence +energy heats the ions (Arzamasskiy et al. 2019). +Re- +cent theoretical work have suggested that conservation +of magnetic helicity prevents the turbulence energy from +cascading toward sub-ion scales, thus most of the cas- +caded energy is absorbed by the ions (Squire et al. 2022). +Bacchini et al. (2022) show that during the transition +from Alfv´en waves to kinetic Alfv´en waves at sub-ion +scales, ions can gain more energy than electrons be- +cause the kinetic energy of the waves is mostly acces- +sible to ions instead of electrons. +In addition to the +wave-particle interaction, other effects such as intermit- +tency (e.g. Osman et al. 2012) and stochastic heating +(e.g. Chandran et al. 2010) may also contribute to the +differential heating process. Sioulas et al. (2022a), us- +ing Parker Solar Probe measurements, show that pro- +tons can gain more energy from the intermittent struc- +tures than the electrons. The stochastic heating is pos- +itively correlated with the turbulence strength (Vech +et al. 2017) and is shown to be significant throughout +the inner heliosphere (Martinovi´c et al. 2019, 2020), con- +tributing to the perpendicular temperature of ions. +In this study, we show that, for an Alfv´en wave driven +solar wind, if most of the wave energy dissipates into +ions, the positive Tp − VSW (Tp is proton temperature + +ion and electron temperatures in the solar wind +3 +Figure 1. Left column: Distribution of data collected below 30 solar radii during first nine orbits of Parker Solar Probe. Right +column: Distribution of one year (2019) of data from WIND. From top to bottom, the rows are proton number density Np, +proton temperature Tp, electron temperature Te, and amplitude of the magnetic field fluctuations |δB| as functions of the radial +solar wind speed VSW . For PSP data, we use one-minute time windows to calculate the average values of the quantities and +the fluctuation strength. For WIND data, we use five-minute time windows. The gray squares show the median values and the +error bars show the root-mean-squares of the binned data. +and VSW is solar wind speed) correlation and negative +Te − VSW (Te is electron temperature) correlation are +naturally generated. The paper is organized as follows: +In Section 2, we present Parker Solar Probe (PSP) and +WIND observations of the solar wind and show the cor- +relations between temperatures and solar wind speed. +In Section 3, we describe the 1D Alfv´en wave driven +solar wind model used in this study and present the +simulation results. In Section 4 we discuss the underly- +ing mechanism that explains the numerical results. In +Section 5 we conclude this study. +2. WIND & PSP OBSERVATIONS +We use PSP and WIND data to investigate the cor- +relation between the solar wind speed and various solar +wind parameters. For PSP, we use data from the first +nine orbits and we only select data collected below 30 so- +lar radii. For proton measurements, we mainly use data +from the electrostatic analyzer (SPAN-Ion) but use the +Faraday cup (SPC) for the first orbit when high-quality +SPAN-Ion data is unavailable (Fox et al. 2016; Kasper +et al. 2016). The electron temperature is the derived +core temperature by fitting the electron velocity distri- +bution functions measured by SPAN-Electron (Halekas +et al. 2020). The time cadence of SPAN data is typically +7-14 sec, and the time cadence of SPC data is around +0.44 sec. +The magnetic field data is collected by the +fluxgate magnetometer with a time cadence of 3.4 mil- +liseconds (Fox et al. 2016; Bale et al. 2016). For WIND, +we use one year of data collected in 2019 and we have +verified that the 2020 data give very similar results. The +proton data is from the 3D Plasma Analyzer (3DP) elec- +trostatic analyzers with three second cadence (Lin et al. +1995), the electron data is from the Solar Wind Experi- +ment (SWE) electron instruments with 6-12 second ca- +dence (Ogilvie et al. 1995), and the magnetic field data is +from the Wind Magnetic Field Investigation (MFI) flux- + +PSP R ≤ 30Rs +WIND 2019 +15 +2000 - +12 +1500 +3 +/cm +1000 +500 +20 +8 +15 +K +05 +10 +5 +2 +2.0 +K +V +6 +1.5 +/105 +/105 +1.0 +4 : +3 +0.5 +80 +3 +09 +lu +I6BI nT +[6B| +20 +0 - +100 +200 +300 +400 +500 +600 +300 +400 +500 +600 +700 +800 +Vsw km/s +Vsw km/s4 +Shi et al. +gate magnetometers with three second cadence (Lepping +et al. 1995). +In Figure 1, we show distribution of PSP data on the +left column and distribution of WIND data on the right +column. From top to bottom rows are proton number +density Np, proton temperature Tp, electron tempera- +ture Te, and amplitude of the magnetic field fluctua- +tions |δB| as functions of the radial solar wind speed +VSW . For PSP data, we use one-minute time windows +to calculate the average values of the quantities and the +fluctuation strength, defined as the root-mean-square +(RMS) of the magnetic field. For WIND data, we use +five-minute time windows. The gray squares show the +median values and the error bars show the RMS of the +binned data. It is clear in both datasets that higher so- +lar wind speed in general corresponds to lower density, +higher proton temperature, and stronger magnetic field +fluctuations. As previously shown in Maksimovic et al. +(2020); Halekas et al. (2020); Salem et al. (2021); Dakeyo +et al. (2022), the electron temperature has a strong neg- +ative correlation with the solar wind speed as observed +by PSP. However, at 1 AU, Te decreases with VSW for +winds faster than about 520 km/s but seems to increase +with VSW for slower wind streams. +3. 1D TWO-TEMPERATURE ALFV´EN WAVE +POWERED SOLAR WIND MODEL +3.1. Model description +We utilize a 1D Alfv´en-wave-driven solar wind model +with different ion (proton) and electron temperatures. +Wave-driven solar wind models have been developed +and widely used to analyze the heating and acceleration +of solar wind (e.g. Cranmer & Van Ballegooijen 2005; +Cranmer et al. 2007; Chandran & Hollweg 2009; Ver- +dini & Velli 2007; Verdini et al. 2009; Lionello et al. 2014; +Shoda et al. 2018; R´eville et al. 2020). While most of +the previous works assume a one-fluid solar wind, some +works have adopted a two-fluid solar wind model with +different proton and electron temperatures (Chandran +et al. 2011; Adhikari et al. 2022). The model used in +the current study is very similar to the one-fluid model +used by R´eville et al. (2020) but with independent ion +and electron temperatures. The model equations are +B(r) = +A0 +A(r)B0 +(1a) +∂ρ +∂t = − 1 +A +∂ +∂r (ρV A) +(1b) +∂V +∂t = − V ∂V +∂r − 1 +ρ +∂ +∂r +� +P + 1 +2ε +� +− GM +r2 +(1c) +∂Pi +∂t = − V ∂Pi +∂r − γi +1 +A +∂ (AV ) +∂r +Pi + (γi − 1)Qi +(1d) +∂Pe +∂t = − V ∂Pe +∂r − γe +1 +A +∂ (AV ) +∂r +Pe + (γe − 1)Qe +(1e) +∂ε+ +∂t = − (V + VA)∂ε+ +∂r − 1 +A +∂ +∂r (A (V + VA)) ε+ +− 1 +2 +1 +A +∂(AV ) +∂r +ε+ + R+ + D+ +(1f) +∂ε− +∂t = − (V − VA)∂ε− +∂r − 1 +A +∂ +∂r (A (V − VA)) ε− +− 1 +2 +1 +A +∂(AV ) +∂r +ε− + R− + D− +(1g) +with P = Pi + Pe and ε = ε+ + ε−. Here, B(r), ρ(r), +V (r), Pi(r), Pe(r) are the radial magnetic field, plasma +density, radial solar wind speed, ion thermal pressure, +and electron thermal pressure respectively. γi,e are the +polytropic indices for ions and electrons respectively. +We consider a spherically symmetric radial flux tube +such that A(r) is the cross section area of the tube +and A0 is the cross section area at the inner bound- +ary. +ε± = +1 +4ρ |z±|2 with z± being the two Els¨asser +variables. Thus ε± represent the energy densities (per +volume) of the outward and inward propagating Alfv´en +waves. VA = B/√µ0ρ is the radial Alfv´en speed. D± +are the dissipation rates of the two wave populations +due to the nonlinear energy cascade, and R± represent +the reflection of the waves due to the inhomogeneity +of the background plasma. Qi and Qe are the heating +terms for ions and electrons, and each of them consist of +three components such that Qi = Qh,i +Qw,i +Qc,i and +Qe = Qh,e + Qw,e + Qc,e. Here Qh,i and Qh,e are the +ad-hoc heating terms that are significant only at very +low altitudes. +Qw,i and Qw,e are the heating of ions +and electrons by the wave dissipation. Qc,i and Qc,e are +the heating terms caused by collisionless electron heat +conduction (Hollweg 1976). +A comment on the treatment of the Alfv´en wave equa- +tions and their coupling to the solar wind is in order. +Though this model has been used before, it represents +a drastic simplification of the true problem, as it writes +the evolution equations directly in terms of the separate +energy densities of outward and inward modes rather +than the general second order moment of the fluctuat- +ing fields. The latter would imply at least four equations +rather than the two for the fluctuating energies, and a + +ion and electron temperatures in the solar wind +5 +Figure 2. Radial profiles of various quantities in two sets of simulations. Left column has Cw = 1, i.e. all the dissipated wave +energy heats the ions. Right column has Cw = 0.8, i.e. 20% of the dissipated wave energy heats the electrons. From top to +bottom rows are solar wind speed, plasma number density, ion temperature, and electron temperature respectively. In each +panel, curves with different colors correspond to different wave amplitudes at the inner boundary. From dark to light, colors +correspond to increasing wave amplitudes of [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] km/s. +generalized Reynolds stress in the solar wind momen- +tum equation rather than the simple fluctuating mag- +netic pressure of equation (1c). The still unresolved dif- +ficulties of this model have been discussed in, e.g., Velli +(1993), while the approximation used above is effective +in the limit of small reflection - necessary to trigger non- +linear interactions but not large enough to require the +full second order moments - that is satisfied by all ex- +cept the lowest frequency fluctuations (corresponding to +periods of several hours to days). +We use a superradially expanding flux tube (Verdini +et al. 2009; Lionello et al. 2014): A(r) = f(r)r2 with +f(r) = fm + f1 exp (−(r − rexp)/σexp) +1 + exp(−(r − rexp)/σexp) +(2) +and f1 = 1 − (fm − 1) exp((rs − rexp)/σexp). Here rs +is the solar radius, rexp = 1.31rs, σexp = 0.51rs. The +above expression leads to f(rs) = 1 and f(+∞) = fm. +If fm = 1 we get f(r) ≡ 1, which is a radially expanding +flux tube. In this study, we set fm = 4, which is a typical +value for fast solar wind (Wang & Sheeley Jr 1990). The +polytropic indices are γi = γe = 5/3 (adiabatic index) +so that both the species cool as Ti,e ∝ r−4/3 without +other heating terms. +The first three terms on the right-hand-side of equa- +tions (1f & 1g) correspond to the Wentzel-Kramers- +Brillouin (WKB) evolution of the wave amplitudes +(Alazraki & Couturier 1971; Belcher 1971; Hollweg +1974). The reflection term is written as +R± = CR × +����(V ∓ VA) ∂ +∂r ln √ρ +���� ε∓ +(3) + +Cw = 1.00 +Cw = 0.80 +600 +600 +V km/s +S +400 +V km/s +400 +200 - +200 +0 +108 +108 +106. +106 +3 +cm +104. +104 +n +n +102. +102 +101 +101 +105K +105K +T 10° +T 10° +101, +101 +105K +105K +人 +100 +100 +100 +101 +102 +100 +101 +102 +rlRs +r/Rs6 +Shi et al. +Figure 3. +Radial profiles of different heating terms (per +unit mass) in the run with Cw = 0.8 and |z+| = 100 km/s +at the inner boundary. Blue curves are the ad-hoc heating, +orange curves are the wave heating, and green curve is the +collisionless heat conduction. Solid curves are heating of ions +and dashed curves are heating of electrons. The minimum +of the collisionless heating is around −6.1 × 105 m2 · s−3 at +r = 1.85Rs. +with CR = 0.1 being a constant coefficient. The nonlin- +ear dissipation is +D± = −1 +8ρ|z∓| |z±|2 +λ += − +√ +ε∓ε± +√ρλ +(4) +where λ(r) is the perpendicular correlation length +(R´eville et al. 2020) and is modeled as +λ(r) = λ0 +� +A(r) +A0 +(5) +In this study, we set λ0 = 6×107m, similar to the typical +size of large supergranules (Verdini & Velli 2007; Verdini +et al. 2009; R´eville et al. 2020). +Since the dissipated +wave energy heats the protons and electrons, we have +Qw,i + Qw,e = −(D+ + D−). +(6) +In the model, a free parameter Cw ∈ [0, 1] controls the +portion of the dissipated wave energy that heats the ions: +Qw,i = −Cw(D+ + D−), Qw,e = −(1 − Cw)(D+ + D−). +(7) +The ad-hoc heating is +Qh,(i,e) = Q0,(i,e) +A0 +A exp +� +−r − rs +rs +� +(8) +and we set Q0,i = 5 × 10−7J · m−3 · s−1 and Q0,e = +2 × 10−7J · m−3 · s−1. These terms represent contribu- +tions from processes such as the nanoflares (e.g. Cargill +& Klimchuk 2004) that are important in the low corona. +We choose Q0,e < Q0,i because remote-sensing obser- +vations reveal that the electron temperature is smaller +than the proton temperature in the coronal holes (Cran- +mer 2009). The collisionless heat conduction term writes +as +Qc,i = 0, Qc,e = − 1 +A +∂ +∂r (Aqc) +(9) +where qc = 3 +2PeV (Hollweg 1976). Note that the colli- +sionless heat conduction takes effect for electrons only. +The simulation domain is r ∈ [1, 215]rs with nonuni- +form grid. +The spatial resolution is ∆x = 0.001rs +at the inner boundary and ∆x = 0.1rs at the outer +boundary, and the total number of grid points is N = +2934. +Dirichlet boundary conditions are imposed for +(B, ρ, Pi, Pe, ε+) at the inner boundary and the outer +boundary is open so the wind and waves can propa- +gate out of the domain freely. We note that no inner +boundary conditions are needed for V and ε− because +the sonic point and Alfv´en point implicitly impose two +constraints for them (Parker 1958; Barkhudarov 1991; +Velli 1993). +In all the simulations, we set B = 5G, +n = 1 × 108cm−3, Ti = Te = 1MK at the inner bound- +ary where n is the number density of the plasma, Ti +and Te are the ion temperature and electron tempera- +ture respectively. We carry out five sets of simulations +with Cw = [1, 0.95, 0.9, 0.85, 0.8]. For each Cw, we do +a series of runs with varying ε+(rs) such that |z+| = +[0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] km/s at +the inner boundary. We run each simulation until all +the fields reach a stationary state (∂t = 0) and acquire +the radial profiles of the fields. +3.2. Results +In Figure 2, we plot the radial profiles of various +quantities in two sets of runs. The left column shows +runs with Cw = 1 and the right column shows runs +with Cw = 0.8. +From top to bottom rows are solar +wind speed, plasma number density, ion temperature, +and electron temperature. In each panel, dark to light +colors correspond to runs with increasing values of the +inner boundary wave amplitude from 0 to 100 km/s. +Larger wave energy input leads to a higher solar wind +speed because of a stronger wave pressure gradient and +more heating of the plasma. The wave amplitude does +not change the density profile much. The left column +clearly shows that larger wave amplitude leads to higher +ion temperature and lower electron temperature. +For +Cw = 0.8, the ion temperature still increases with the +wave amplitude, while the behavior of electron temper- +ature is more complicated. Close to the Sun (r ≲ 20rs), +electron temperature decreases with wave amplitude. + +Cw = 0.80, |z|= 100 km/s +108 +ad-hoc +107 +wave +collisionless +106 +ion +105 +electron +S +104. +103 +d/o +102. +101. +100. +0 +100 +101 +102 +r/Rsion and electron temperatures in the solar wind +7 +Figure 4. Plasma number density (top), ion temperature (bottom blue), and electron temperature (bottom orange) as functions +of the solar wind speed at 1 AU. From left to right panels are runs corresponding to Cw =1, 0.9, and 0.8 respectively. +Figure 5. Ti − V (top) and Te − V (bottom) correlations at different radial distances to the Sun in runs with Cw =1 (left), 0.9 +(middle), and 0.8 (right). In each panel, from dark to light colors correspond to r =10, 25, 50, 100, and 215 solar radii. +Further away from the Sun, electron temperature in- +creases with the wave amplitude. Figure 3 shows contri- +butions to ion heating (solid curves) and electron heat- +ing (dashed curves) per unit mass, i.e. Q/ρ, by different +mechanisms in the run with Cw = 0.8 and |z+| = 100 +km/s at the inner boundary. Blue curves are the ad- +hoc heating, orange curves are the wave heating, and +the green curve is the collisionless electron thermal con- +duction. The wave heating is weaker than the ad-hoc +heating close to the Sun, but the radial extent of sig- +nificant wave heating is larger than the ad-hoc heating. +In addition, the collisionless thermal conduction is com- +parable or even larger than wave heating far away from +the Sun. +In Figure 4, we show how the plasma number density +(top panel), ion temperature (bottom blue) and elec- +tron temperature (bottom orange) vary with the solar +wind speed at 1 AU in runs with different Cw. From left +to right columns are Cw = 1, 0.9, and 0.8 respectively. +The behavior of plasma density does not depend on Cw + +Cw = 1.00 +Cw = 0.90 +Cw = 0.80 +14.5 +14.5 +14.5 +14.0 +14.0 +14.0 +3 +13.5 +13.5 +AU) +13.0 +13.0 +12.5 +12.5 +12.5 +12.0 +12.0 +12.0 +0.8 +0.8 +0.8 +105k +0.6 +0.6 +0.6 +0.4 +0.4 +0.2 +Te +0.2 +0.2 +450 +500 +550 +600 +650 +450 +500 +550 +600 +650 +450 +500 +550 +600 +650 +V(1AU) km/s +V(1AU) km/s +V(1AU) km/sCw = 0.80 +H +101 +101 +101 +105K +100 +100- +100 +6×100 +4×100 +105K +3×100 +2 ×100 +100. +100 +100 +400 +500 +600 +700 +400 +500 +600 +700 +400 +500 +600 +700 +V km/s +V km/s +V km/s8 +Shi et al. +Figure 6. Ti − V (top) and Te − V (bottom) correlations +at 1AU for three sets of runs with different amplitudes of +electron ad-hoc heating. Blue curves are Q0,e = 2 × 10−7 +J/m3/s, orange curves are Q0,e = 1 × 10−7 J/m3/s, and +green curves are Q0,e = 0. All the runs have Cw = 1. +as how the dissipated wave energy is distributed among +ions and electrons does not affect the radial profile of +solar wind speed or the density much. As we increase +the wave amplitude, the density drops at first and then +starts to increase, though only with small variation (n +varies between 12.5 and 14.5 cm−3). +Since the mass +conservation law gives n(r) = n0 × (A0V0/A(r)V (r)), +as the wave amplitude increases, if V0 increases slower +than V (r), n(r) has an anti-correlation with V (r), and +vice versa. Hence, the small variation of density with +wave amplitude indicates that the waves modify V0 and +V (r) with similar proportions. Similar to the density, +ion temperature is not modified by Cw significantly, ei- +ther. On the contrary, the electron temperature is quite +sensitive to Cw. As Cw decreases, the negative Te − V +correlation gradually turns to positive, consistent with +what is shown by Figure 2. +In Figure 5, we show Ti − V (top row) and Te − V +(bottom row) at different radial distances to the Sun +for runs with Cw = 1 (left), Cw = 0.9 (middle), and +Cw = 0.8 (right) respectively. In each panel, from dark +to light colors correspond to r = 10, 25, 50, 100, and +215 solar radii. For Cw = 1, positive Ti − V correlation +and negative Te − V correlation are well established at +very close distance to the Sun and maintained as the +wind propagates. +For Cw = 0.9 and Cw = 0.8, the +ion temperature is not modified much, while the Te − V +correlation evolves as the solar wind propagates. Close +to the Sun, negative Te − V correlation is produced, +while as r increases, Te − V correlation gradually turns +positive. This trend is similar to in-situ measurements +by multiple satellites (Maksimovic et al. 2020). +4. DISCUSSION +The positive Ti − V correlation is easy to under- +stand: With more wave energy injected from the in- +ner boundary, the solar wind speed increases because of +larger wave pressure and larger thermal pressure gradi- +ent. Meanwhile, because most of the dissipated wave en- +ergy heats the ions, the ion temperature also increases, +resulting in a positive Ti − V correlation. The cause of +negative Te − V correlation is more complicated. If we +consider the most simple case where the electron fluid +is polytropic such that Te(r) = Te0 × (ρ0/ρ(r))γ−1, the +Te − V relation should be similar to n − V relation. +However, the left column of Figure 4 shows that even +in the large-V regime where the density increases with +V , Te still decreases with V . This indicates that the ad- +hoc heating may play an important role in forming the +Te − V anti-correlation. By observing the equation for +electron pressure (equation (1e)), we see that, in a sta- +tionary state (∂t = 0) and without Alfv´en wave heating +of the electrons, there is +∂Pe +∂r = C1Pe +1 +A +∂A +∂r + C2Pe +1 +V +∂V +∂r + C3 +Qh,e +V +(10) +where C1, C2, and C3 are constants depending on γe +and the collisionless heat conduction strength. Close to +the inner boundary, the dominating term is the ad-hoc +heating term (Figure 3). Because Qh,e(r) is a given func- +tion of r, the contribution of this term to the increment +of pressure is inversely proportional to the solar wind +speed. That is to say, the faster the plasma is ejected, +the less internal energy it gains during its propagation. +This is why a stronger Alfv´en wave injection leads to a +lower electron temperature at close distances to the Sun. +In Figure 6, we show Ti − V (top) and Te − V (bottom) +relations at 1AU for three sets of runs with Cw = 1 +and varying amplitudes of electron ad-hoc heating. The +blue curves correspond to Q0,e = 2 × 10−7 J/m3/s, the +orange curves correspond to Q0,e = 1 × 10−7 J/m3/s, +and the green curves correspond to Q0,e = 0. One can +see that, as we decrease Q0,e, the negative Te − V cor- +relation gradually vanishes, implying that the ad-hoc + +0.9 +0.8 - +0.7 +Ti(1AU) 105K +0.6 +0.5 +0.4 +0.3 - +0.2 +0.1 +Qo,e = 2 × 10-7 J/m3/s +Qo,e = 1 × 10-7 J/m3/s +0.8 - +Qo,e= 0 +K +0.6 +0.4 +0.2 +0.0 +350 +400 +450 +500 +550 +600 +650 +V(1AU) km/sion and electron temperatures in the solar wind +9 +heating is necessary for the negative Te − V correla- +tion. +However, if the electrons gain a portion of the +wave energy during the solar wind expansion, the anti- +correlation between Te and V is gradually destroyed, be- +cause further away from the Sun the contribution of the +ad-hoc heating gradually becomes less important com- +pared with the contribution of the wave dissipation (Fig- +ure 3). This explains why there is a radial evolution of +the Te − V relation. Based on this scenario, the in-situ +observations (Figure 1) indicate that in the slow solar +wind, electrons get more heating during the solar wind +propagation compared with electrons in the fast solar +wind, leading to different Te − V correlations for slow +and fast streams (right column of Figure 1). The under- +lying mechanisms, however, need further studies. +5. CONCLUSION +Through a 1D Alfv´en-wave-driven solar wind model +with different ion and electron temperatures, we have +successfully reproduced the positive correlation between +the ion temperature (Ti) and solar wind speed (V ) and +the negative correlation between the electron tempera- +ture (Te) and solar wind which are observed in the solar +wind. In our simulations, the different Ti−V and Te−V +relations are a result of the fact that most of the dissi- +pated Alfv´en wave energy heats ions instead of electrons +(e.g. Arzamasskiy et al. 2019; Squire et al. 2022; Bac- +chini et al. 2022), making electron heating close to the +Sun by the ad-hoc heating term, which represents mech- +anisms such as magnetic reconnection, less efficient due +to faster wind speed. With a small but finite portion +of the dissipated wave energy heating the electrons, the +simulations also reproduce the observed radial evolution +of the Te − V relation, i.e., the initially negative corre- +lation gradually turns into a positive one (Maksimovic +et al. 2020), because the contribution of the ad-hoc heat- +ing term gradually becomes negligible compared with +the Alfv´en wave heating as the wind propagates. +We note that the model used here is not fully self- +consistent and some important characteristics of the ob- +served solar wind are missing in the model results. First, +the temperature evolution given by the model only qual- +itatively, but not completely quantitatively, agrees with +the observations. The radial decay rate (α in Te ∝ r−α) +of the electron temperature is large and does not vary +much with solar wind speed. In the bottom-right panel +of Figure 2, α changes from ∼ 0.67 to ∼ 0.65 as the wind +speed at 1AU increases from 470km/s to 660km/s. In +contrast, α varies between ∼ 0.4 and ∼ 0.2 as wind speed +changes from 400km/s to 600km/s as estimated using +HELIOS data (Maksimovic et al. 2020). +This differ- +ence implies that other mechanisms, e.g. electron heat +flux, omitted in our model play an important role in lo- +cal electron heating. Second, the density only changes +slightly among runs with different wave amplitudes (Fig- +ure 2) and thus density does not vary with the wind +speed significantly (top row of Figure 4). However, it is +well known that faster solar wind is generally less dense +than slower solar wind and the mass flux only moder- +ately depends on the solar wind speed (Wang 2010). +Both the inner boundary plasma density and the ex- +pansion factor can heavily affect the solar wind density, +but they are both constant in the current study. A thor- +ough parametric study in which all these parameters are +treated as variables is necessary and will be conducted in +the future. 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H. 2011, Journal of +Geophysical Research: Space Physics, 116 +Wang, Y.-M. 2010, The Astrophysical Journal Letters, 715, +L121 +Wang, Y.-M., & Sheeley Jr, N. 1990, The Astrophysical +Journal, 355, 726 + diff --git a/StAyT4oBgHgl3EQf8PpF/content/tmp_files/load_file.txt b/StAyT4oBgHgl3EQf8PpF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d67b2b6d707564aee5bb3b4684a4f5bd9a9c35f8 --- /dev/null +++ b/StAyT4oBgHgl3EQf8PpF/content/tmp_files/load_file.txt @@ -0,0 +1,879 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf,len=878 +page_content='Draft version January 4, 2023 Typeset using LATEX twocolumn style in AASTeX631 Proton and electron temperatures in the solar wind and their correlations with the solar wind speed Chen Shi (时辰) ,1 Marco Velli ,1 Roberto Lionello ,2 Nikos Sioulas ,1 Zesen Huang (黄泽森) ,1 Jasper S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' CNES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' France 6LESIA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Observatoire de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Universit e PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Sorbonne Universit e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Universit e de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 5 place Jules Janssen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 92195 Meudon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' France 7Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' CA 94720-7300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' USA 8Space Sciences Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' CA 94720-7450,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' USA ABSTRACT The heating and acceleration of the solar wind remains one of the fundamental unsolved problems in heliophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' It is usually observed that the proton temperature Ti is highly correlated with the solar wind speed VSW , while the electron temperature Te shows anti-correlation or no clear correlation with the solar wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Here we inspect both Parker Solar Probe (PSP) and WIND data and compare the observations with simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' PSP observations below 30 solar radii clearly show a positive correlation between proton temperature and wind speed and a negative correlation between electron temperature and wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' One year (2019) of WIND data confirm that proton temperature is positively correlated with solar wind speed, but the electron temperature increases with the solar wind speed for slow wind while it decreases with the solar wind speed for fast wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Using a one-dimensional Alfv´en-wave-driven solar wind model with different proton and electron temperatures, we for the first time find that if most of the dissipated Alfv´en wave energy heats the ions instead of electrons, a positive Ti − VSW correlation and a negative Te − VSW correlation arise naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' If the electrons gain a small but finite portion of the dissipated wave energy, the Te − VSW correlation evolves with radial distance to the Sun such that the negative correlation gradually turns positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The model results show that Alfv´en waves are one of the possible explanations of the observed evolution of proton and electron temperatures in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Keywords: Magnetohydrodynamics (1964), Solar wind (1534), Alfven waves (23) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' INTRODUCTION Solar wind is the plasma flow ejected from solar corona, filling the interplanetary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' It carries a large amount of mass and energy out of the Sun and serves Corresponding author: Chen Shi cshi1993@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='edu as the medium of various physical processes and struc- tures, such as waves and turbulence, coronal mass ejec- tion, and magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Solar wind continu- ously interacts with the Earth and injects energy in the Earth’s magnetosphere, causing strong disturbances of the Earth’s magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Understanding the gener- ation of solar wind and the dynamics of the plasma in the solar wind is necessary not only for a better space arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='00852v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='SR] 2 Jan 2023 ID2 Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' weather prediction but also for a deeper insight of the fundamental plasma astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' A significant amount of energy is needed for the plasma to escape the solar gravity and to accelerate to a supersonic speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Early works (Parker 1958, 1964a,b, 1965) show that an isothermal solar corona with strong thermal conduction can generate the ob- served solar wind speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' However, in-situ measure- ments imply that in the inner heliosphere both ion and electron temperatures of the solar wind decay with ra- dial distance to the Sun and the radial profiles of the temperatures can be fitted with polytropic relations (Marsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' ˇStver´ak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Boldyrev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The poly- tropic indices deduced from the in-situ measurements are in general smaller than the adiabatic index 5/3, im- plying that in-situ heating mechanisms are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' One possible source of the in-situ heating is the tur- bulence energy cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In the solar wind, predomi- nantly outward propagating Alfv´en waves exist (Belcher & Davis Jr 1971), while smaller-amplitude inward prop- agating Alfv´en waves are generated due to processes in- cluding wave reflection induced by the gradient of Alfv´en speed (Heinemann & Olbert 1980) and the large-scale stream shears (Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The nonlinear interaction between the outward and inward propagating waves leads to a turbulence energy cascade (Kraichnan 1965), which eventually dissipates the en- ergy of the electromagnetic fields into the plasma energy through wave-particle interactions (Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2017) and intermittent structures (Os- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Matthaeus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Sioulas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' It has been long observed that in the solar wind, the proton temperature has a strong positive correla- tion with the solar wind speed (Burlaga & Ogilvie 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Lopez & Freeman 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Matthaeus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' D´emoulin 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Elliott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Hofmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This positive correlation indicates the possibility that a particular mechanism contributes to the proton heating and solar wind momentum simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' On the contrary, electron core temperature usually shows anti-correlation with the solar wind speed (Marsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020) and this anti-correlation may originate at the source of so- lar wind due to interchange reconnection (Fisk 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Gloeckler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2003) as the coronal electron temper- ature estimated from the fractions between heavy ions measured in-situ, which is a good proxy for freeze-in temperatures of the solar corona, has an anti-correlation with solar wind speed (Geiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Ko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Gloeckler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' von Steiger & Zurbuchen 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Here, we propose a mechanism that the different temperature-speed correlations for protons and elec- trons are possibly generated in-situ as the solar wind propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The major factor is how the energy source for the acceleration of solar wind deposits differently into protons and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' As the turbulence energy cascades from MHD inertial scales toward ion kinetic scales, various kinetic processes may arise and they determine how the electromagnetic energy eventually heats the ions and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Two major processes are kinetic Alfv´en waves (KAWs) which mainly heat the electrons through Landau damping, and ion cyclotron waves (ICWs) which heat the ions through cyclotron resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Both KAWs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Podesta 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2012) and ICWs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Jian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2010, 2009) are identi- fied in the solar wind, and recent observation made by Parker Solar Probe shows that the two modes may coex- ist (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Gyrokinetic theory and simula- tions show that the ratio between ion heating and elec- tron heating during dissipation of Alfv´enic turbulence is positively correlated with ion beta (ratio between ion thermal pressure and magnetic pressure) (Howes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Schekochihin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Howes 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Kawazura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Moreover, existence of compressive compo- nent in the turbulence will increase the ratio between ion and electron heating (Kawazura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Hybrid- kinetic simulations using realistic solar wind parameters at 1 AU show that 75-80% of the cascaded turbulence energy heats the ions (Arzamasskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Re- cent theoretical work have suggested that conservation of magnetic helicity prevents the turbulence energy from cascading toward sub-ion scales, thus most of the cas- caded energy is absorbed by the ions (Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2022) show that during the transition from Alfv´en waves to kinetic Alfv´en waves at sub-ion scales, ions can gain more energy than electrons be- cause the kinetic energy of the waves is mostly acces- sible to ions instead of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In addition to the wave-particle interaction, other effects such as intermit- tency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Osman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2012) and stochastic heating (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Chandran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2010) may also contribute to the differential heating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Sioulas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2022a), us- ing Parker Solar Probe measurements, show that pro- tons can gain more energy from the intermittent struc- tures than the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The stochastic heating is pos- itively correlated with the turbulence strength (Vech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2017) and is shown to be significant throughout the inner heliosphere (Martinovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2019, 2020), con- tributing to the perpendicular temperature of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In this study, we show that, for an Alfv´en wave driven solar wind, if most of the wave energy dissipates into ions, the positive Tp − VSW (Tp is proton temperature ion and electron temperatures in the solar wind 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Left column: Distribution of data collected below 30 solar radii during first nine orbits of Parker Solar Probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Right column: Distribution of one year (2019) of data from WIND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From top to bottom, the rows are proton number density Np, proton temperature Tp, electron temperature Te, and amplitude of the magnetic field fluctuations |δB| as functions of the radial solar wind speed VSW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For PSP data, we use one-minute time windows to calculate the average values of the quantities and the fluctuation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For WIND data, we use five-minute time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The gray squares show the median values and the error bars show the root-mean-squares of the binned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' and VSW is solar wind speed) correlation and negative Te − VSW (Te is electron temperature) correlation are naturally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The paper is organized as follows: In Section 2, we present Parker Solar Probe (PSP) and WIND observations of the solar wind and show the cor- relations between temperatures and solar wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Section 3, we describe the 1D Alfv´en wave driven solar wind model used in this study and present the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Section 4 we discuss the underly- ing mechanism that explains the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Section 5 we conclude this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' WIND & PSP OBSERVATIONS We use PSP and WIND data to investigate the cor- relation between the solar wind speed and various solar wind parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For PSP, we use data from the first nine orbits and we only select data collected below 30 so- lar radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For proton measurements, we mainly use data from the electrostatic analyzer (SPAN-Ion) but use the Faraday cup (SPC) for the first orbit when high-quality SPAN-Ion data is unavailable (Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The electron temperature is the derived core temperature by fitting the electron velocity distri- bution functions measured by SPAN-Electron (Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The time cadence of SPAN data is typically 7-14 sec, and the time cadence of SPC data is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='44 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The magnetic field data is collected by the fluxgate magnetometer with a time cadence of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='4 mil- liseconds (Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For WIND, we use one year of data collected in 2019 and we have verified that the 2020 data give very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The proton data is from the 3D Plasma Analyzer (3DP) elec- trostatic analyzers with three second cadence (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1995), the electron data is from the Solar Wind Experi- ment (SWE) electron instruments with 6-12 second ca- dence (Ogilvie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1995), and the magnetic field data is from the Wind Magnetic Field Investigation (MFI) flux- PSP R ≤ 30Rs WIND 2019 15 2000 - 12 1500 3 /cm 1000 500 20 8 15 K 05 10 5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 K V 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 /105 /105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 4 : 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 80 3 09 lu I6BI nT [6B| 20 0 - 100 200 300 400 500 600 300 400 500 600 700 800 Vsw km/s Vsw km/s4 Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' gate magnetometers with three second cadence (Lepping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Figure 1, we show distribution of PSP data on the left column and distribution of WIND data on the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From top to bottom rows are proton number density Np, proton temperature Tp, electron tempera- ture Te, and amplitude of the magnetic field fluctua- tions |δB| as functions of the radial solar wind speed VSW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For PSP data, we use one-minute time windows to calculate the average values of the quantities and the fluctuation strength, defined as the root-mean-square (RMS) of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For WIND data, we use five-minute time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The gray squares show the median values and the error bars show the RMS of the binned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' It is clear in both datasets that higher so- lar wind speed in general corresponds to lower density, higher proton temperature, and stronger magnetic field fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' As previously shown in Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Dakeyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2022), the electron temperature has a strong neg- ative correlation with the solar wind speed as observed by PSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' However, at 1 AU, Te decreases with VSW for winds faster than about 520 km/s but seems to increase with VSW for slower wind streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 1D TWO-TEMPERATURE ALFV´EN WAVE POWERED SOLAR WIND MODEL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Model description We utilize a 1D Alfv´en-wave-driven solar wind model with different ion (proton) and electron temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Wave-driven solar wind models have been developed and widely used to analyze the heating and acceleration of solar wind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Cranmer & Van Ballegooijen 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Chandran & Hollweg 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Ver- dini & Velli 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Verdini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Lionello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Shoda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' R´eville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' While most of the previous works assume a one-fluid solar wind, some works have adopted a two-fluid solar wind model with different proton and electron temperatures (Chandran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Adhikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The model used in the current study is very similar to the one-fluid model used by R´eville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (2020) but with independent ion and electron temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The model equations are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='B(r) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A(r)B0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂t = − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r (ρV A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂t = − V ∂V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='P + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='− GM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂Pi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂t = − V ∂Pi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r − γi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ (AV ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='Pi + (γi − 1)Qi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂Pe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂t = − V ∂Pe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r − γe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ (AV ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='Pe + (γe − 1)Qe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ε+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂t = − (V + VA)∂ε+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r (A (V + VA)) ε+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂(AV ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='ε+ + R+ + D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ε− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂t = − (V − VA)∂ε− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r (A (V − VA)) ε− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂(AV ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='∂r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='ε− + R− + D− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='(1g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='with P = Pi + Pe and ε = ε+ + ε−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Here, B(r), ρ(r), V (r), Pi(r), Pe(r) are the radial magnetic field, plasma density, radial solar wind speed, ion thermal pressure, and electron thermal pressure respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' γi,e are the polytropic indices for ions and electrons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We consider a spherically symmetric radial flux tube such that A(r) is the cross section area of the tube and A0 is the cross section area at the inner bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' ε± = 1 4ρ |z±|2 with z± being the two Els¨asser variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Thus ε± represent the energy densities (per volume) of the outward and inward propagating Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' VA = B/√µ0ρ is the radial Alfv´en speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' D± are the dissipation rates of the two wave populations due to the nonlinear energy cascade, and R± represent the reflection of the waves due to the inhomogeneity of the background plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Qi and Qe are the heating terms for ions and electrons, and each of them consist of three components such that Qi = Qh,i +Qw,i +Qc,i and Qe = Qh,e + Qw,e + Qc,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Here Qh,i and Qh,e are the ad-hoc heating terms that are significant only at very low altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Qw,i and Qw,e are the heating of ions and electrons by the wave dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Qc,i and Qc,e are the heating terms caused by collisionless electron heat conduction (Hollweg 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' A comment on the treatment of the Alfv´en wave equa- tions and their coupling to the solar wind is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Though this model has been used before, it represents a drastic simplification of the true problem, as it writes the evolution equations directly in terms of the separate energy densities of outward and inward modes rather than the general second order moment of the fluctuat- ing fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The latter would imply at least four equations rather than the two for the fluctuating energies, and a ion and electron temperatures in the solar wind 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Radial profiles of various quantities in two sets of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Left column has Cw = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' all the dissipated wave energy heats the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Right column has Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 20% of the dissipated wave energy heats the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From top to bottom rows are solar wind speed, plasma number density, ion temperature, and electron temperature respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In each panel, curves with different colors correspond to different wave amplitudes at the inner boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From dark to light, colors correspond to increasing wave amplitudes of [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' generalized Reynolds stress in the solar wind momen- tum equation rather than the simple fluctuating mag- netic pressure of equation (1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The still unresolved dif- ficulties of this model have been discussed in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=', Velli (1993), while the approximation used above is effective in the limit of small reflection - necessary to trigger non- linear interactions but not large enough to require the full second order moments - that is satisfied by all ex- cept the lowest frequency fluctuations (corresponding to periods of several hours to days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We use a superradially expanding flux tube (Verdini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Lionello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2014): A(r) = f(r)r2 with f(r) = fm + f1 exp (−(r − rexp)/σexp) 1 + exp(−(r − rexp)/σexp) (2) and f1 = 1 − (fm − 1) exp((rs − rexp)/σexp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Here rs is the solar radius, rexp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='31rs, σexp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='51rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The above expression leads to f(rs) = 1 and f(+∞) = fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' If fm = 1 we get f(r) ≡ 1, which is a radially expanding flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In this study, we set fm = 4, which is a typical value for fast solar wind (Wang & Sheeley Jr 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The polytropic indices are γi = γe = 5/3 (adiabatic index) so that both the species cool as Ti,e ∝ r−4/3 without other heating terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The first three terms on the right-hand-side of equa- tions (1f & 1g) correspond to the Wentzel-Kramers- Brillouin (WKB) evolution of the wave amplitudes (Alazraki & Couturier 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Belcher 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Hollweg 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The reflection term is written as R± = CR × ����(V ∓ VA) ∂ ∂r ln √ρ ���� ε∓ (3) Cw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='00 Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='80 600 600 V km/s S 400 V km/s 400 200 - 200 0 108 108 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 106 3 cm 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 104 n n 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 102 101 101 105K 105K T 10° T 10° 101, 101 105K 105K 人 100 100 100 101 102 100 101 102 rlRs r/Rs6 Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Radial profiles of different heating terms (per unit mass) in the run with Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 and |z+| = 100 km/s at the inner boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Blue curves are the ad-hoc heating, orange curves are the wave heating, and green curve is the collisionless heat conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Solid curves are heating of ions and dashed curves are heating of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The minimum of the collisionless heating is around −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 × 105 m2 · s−3 at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='85Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' with CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 being a constant coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The nonlin- ear dissipation is D± = −1 8ρ|z∓| |z±|2 λ = − √ ε∓ε± √ρλ (4) where λ(r) is the perpendicular correlation length (R´eville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020) and is modeled as λ(r) = λ0 � A(r) A0 (5) In this study, we set λ0 = 6×107m, similar to the typical size of large supergranules (Verdini & Velli 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Verdini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' R´eville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Since the dissipated wave energy heats the protons and electrons, we have Qw,i + Qw,e = −(D+ + D−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (6) In the model, a free parameter Cw ∈ [0, 1] controls the portion of the dissipated wave energy that heats the ions: Qw,i = −Cw(D+ + D−), Qw,e = −(1 − Cw)(D+ + D−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' (7) The ad-hoc heating is Qh,(i,e) = Q0,(i,e) A0 A exp � −r − rs rs � (8) and we set Q0,i = 5 × 10−7J · m−3 · s−1 and Q0,e = 2 × 10−7J · m−3 · s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' These terms represent contribu- tions from processes such as the nanoflares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Cargill & Klimchuk 2004) that are important in the low corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We choose Q0,e < Q0,i because remote-sensing obser- vations reveal that the electron temperature is smaller than the proton temperature in the coronal holes (Cran- mer 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The collisionless heat conduction term writes as Qc,i = 0, Qc,e = − 1 A ∂ ∂r (Aqc) (9) where qc = 3 2PeV (Hollweg 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Note that the colli- sionless heat conduction takes effect for electrons only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The simulation domain is r ∈ [1, 215]rs with nonuni- form grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The spatial resolution is ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='001rs at the inner boundary and ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1rs at the outer boundary, and the total number of grid points is N = 2934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Dirichlet boundary conditions are imposed for (B, ρ, Pi, Pe, ε+) at the inner boundary and the outer boundary is open so the wind and waves can propa- gate out of the domain freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We note that no inner boundary conditions are needed for V and ε− because the sonic point and Alfv´en point implicitly impose two constraints for them (Parker 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Barkhudarov 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Velli 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In all the simulations, we set B = 5G, n = 1 × 108cm−3, Ti = Te = 1MK at the inner bound- ary where n is the number density of the plasma, Ti and Te are the ion temperature and electron tempera- ture respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We carry out five sets of simulations with Cw = [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For each Cw, we do a series of runs with varying ε+(rs) such that |z+| = [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] km/s at the inner boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We run each simulation until all the fields reach a stationary state (∂t = 0) and acquire the radial profiles of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Results In Figure 2, we plot the radial profiles of various quantities in two sets of runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The left column shows runs with Cw = 1 and the right column shows runs with Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From top to bottom rows are solar wind speed, plasma number density, ion temperature, and electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In each panel, dark to light colors correspond to runs with increasing values of the inner boundary wave amplitude from 0 to 100 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Larger wave energy input leads to a higher solar wind speed because of a stronger wave pressure gradient and more heating of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The wave amplitude does not change the density profile much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The left column clearly shows that larger wave amplitude leads to higher ion temperature and lower electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8, the ion temperature still increases with the wave amplitude, while the behavior of electron temper- ature is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Close to the Sun (r ≲ 20rs), electron temperature decreases with wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='80, |z|= 100 km/s 108 ad-hoc 107 wave collisionless 106 ion 105 electron S 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 103 d/o 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 0 100 101 102 r/Rsion and electron temperatures in the solar wind 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Plasma number density (top), ion temperature (bottom blue), and electron temperature (bottom orange) as functions of the solar wind speed at 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From left to right panels are runs corresponding to Cw =1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Ti − V (top) and Te − V (bottom) correlations at different radial distances to the Sun in runs with Cw =1 (left), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9 (middle), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In each panel, from dark to light colors correspond to r =10, 25, 50, 100, and 215 solar radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Further away from the Sun, electron temperature in- creases with the wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Figure 3 shows contri- butions to ion heating (solid curves) and electron heat- ing (dashed curves) per unit mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Q/ρ, by different mechanisms in the run with Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 and |z+| = 100 km/s at the inner boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Blue curves are the ad- hoc heating, orange curves are the wave heating, and the green curve is the collisionless electron thermal con- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The wave heating is weaker than the ad-hoc heating close to the Sun, but the radial extent of sig- nificant wave heating is larger than the ad-hoc heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In addition, the collisionless thermal conduction is com- parable or even larger than wave heating far away from the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Figure 4, we show how the plasma number density (top panel), ion temperature (bottom blue) and elec- tron temperature (bottom orange) vary with the solar wind speed at 1 AU in runs with different Cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' From left to right columns are Cw = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The behavior of plasma density does not depend on Cw Cw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='00 Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='90 Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='80 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 AU) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 105k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 Te 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 450 500 550 600 650 450 500 550 600 650 450 500 550 600 650 V(1AU) km/s V(1AU) km/s V(1AU) km/sCw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='80 H 101 101 101 105K 100 100- 100 6×100 4×100 105K 3×100 2 ×100 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 100 100 400 500 600 700 400 500 600 700 400 500 600 700 V km/s V km/s V km/s8 Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Ti − V (top) and Te − V (bottom) correlations at 1AU for three sets of runs with different amplitudes of electron ad-hoc heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Blue curves are Q0,e = 2 × 10−7 J/m3/s, orange curves are Q0,e = 1 × 10−7 J/m3/s, and green curves are Q0,e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' All the runs have Cw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' as how the dissipated wave energy is distributed among ions and electrons does not affect the radial profile of solar wind speed or the density much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' As we increase the wave amplitude, the density drops at first and then starts to increase, though only with small variation (n varies between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Since the mass conservation law gives n(r) = n0 × (A0V0/A(r)V (r)), as the wave amplitude increases, if V0 increases slower than V (r), n(r) has an anti-correlation with V (r), and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Hence, the small variation of density with wave amplitude indicates that the waves modify V0 and V (r) with similar proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Similar to the density, ion temperature is not modified by Cw significantly, ei- ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' On the contrary, the electron temperature is quite sensitive to Cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' As Cw decreases, the negative Te − V correlation gradually turns to positive, consistent with what is shown by Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Figure 5, we show Ti − V (top row) and Te − V (bottom row) at different radial distances to the Sun for runs with Cw = 1 (left), Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9 (middle), and Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 (right) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In each panel, from dark to light colors correspond to r = 10, 25, 50, 100, and 215 solar radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For Cw = 1, positive Ti − V correlation and negative Te − V correlation are well established at very close distance to the Sun and maintained as the wind propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' For Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9 and Cw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8, the ion temperature is not modified much, while the Te − V correlation evolves as the solar wind propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Close to the Sun, negative Te − V correlation is produced, while as r increases, Te − V correlation gradually turns positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This trend is similar to in-situ measurements by multiple satellites (Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' DISCUSSION The positive Ti − V correlation is easy to under- stand: With more wave energy injected from the in- ner boundary, the solar wind speed increases because of larger wave pressure and larger thermal pressure gradi- ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Meanwhile, because most of the dissipated wave en- ergy heats the ions, the ion temperature also increases, resulting in a positive Ti − V correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The cause of negative Te − V correlation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' If we consider the most simple case where the electron fluid is polytropic such that Te(r) = Te0 × (ρ0/ρ(r))γ−1, the Te − V relation should be similar to n − V relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' However, the left column of Figure 4 shows that even in the large-V regime where the density increases with V , Te still decreases with V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This indicates that the ad- hoc heating may play an important role in forming the Te − V anti-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' By observing the equation for electron pressure (equation (1e)), we see that, in a sta- tionary state (∂t = 0) and without Alfv´en wave heating of the electrons, there is ∂Pe ∂r = C1Pe 1 A ∂A ∂r + C2Pe 1 V ∂V ∂r + C3 Qh,e V (10) where C1, C2, and C3 are constants depending on γe and the collisionless heat conduction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Close to the inner boundary, the dominating term is the ad-hoc heating term (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Because Qh,e(r) is a given func- tion of r, the contribution of this term to the increment of pressure is inversely proportional to the solar wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' That is to say, the faster the plasma is ejected, the less internal energy it gains during its propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This is why a stronger Alfv´en wave injection leads to a lower electron temperature at close distances to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In Figure 6, we show Ti − V (top) and Te − V (bottom) relations at 1AU for three sets of runs with Cw = 1 and varying amplitudes of electron ad-hoc heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The blue curves correspond to Q0,e = 2 × 10−7 J/m3/s, the orange curves correspond to Q0,e = 1 × 10−7 J/m3/s, and the green curves correspond to Q0,e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' One can see that, as we decrease Q0,e, the negative Te − V cor- relation gradually vanishes, implying that the ad-hoc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='7 Ti(1AU) 105K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='1 Qo,e = 2 × 10-7 J/m3/s Qo,e = 1 × 10-7 J/m3/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='8 - Qo,e= 0 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='0 350 400 450 500 550 600 650 V(1AU) km/sion and electron temperatures in the solar wind 9 heating is necessary for the negative Te − V correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' However, if the electrons gain a portion of the wave energy during the solar wind expansion, the anti- correlation between Te and V is gradually destroyed, be- cause further away from the Sun the contribution of the ad-hoc heating gradually becomes less important com- pared with the contribution of the wave dissipation (Fig- ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This explains why there is a radial evolution of the Te − V relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Based on this scenario, the in-situ observations (Figure 1) indicate that in the slow solar wind, electrons get more heating during the solar wind propagation compared with electrons in the fast solar wind, leading to different Te − V correlations for slow and fast streams (right column of Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The under- lying mechanisms, however, need further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' CONCLUSION Through a 1D Alfv´en-wave-driven solar wind model with different ion and electron temperatures, we have successfully reproduced the positive correlation between the ion temperature (Ti) and solar wind speed (V ) and the negative correlation between the electron tempera- ture (Te) and solar wind which are observed in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In our simulations, the different Ti−V and Te−V relations are a result of the fact that most of the dissi- pated Alfv´en wave energy heats ions instead of electrons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Arzamasskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Squire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Bac- chini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2022), making electron heating close to the Sun by the ad-hoc heating term, which represents mech- anisms such as magnetic reconnection, less efficient due to faster wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' With a small but finite portion of the dissipated wave energy heating the electrons, the simulations also reproduce the observed radial evolution of the Te − V relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=', the initially negative corre- lation gradually turns into a positive one (Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020), because the contribution of the ad-hoc heat- ing term gradually becomes negligible compared with the Alfv´en wave heating as the wind propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We note that the model used here is not fully self- consistent and some important characteristics of the ob- served solar wind are missing in the model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' First, the temperature evolution given by the model only qual- itatively, but not completely quantitatively, agrees with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The radial decay rate (α in Te ∝ r−α) of the electron temperature is large and does not vary much with solar wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In the bottom-right panel of Figure 2, α changes from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='67 to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='65 as the wind speed at 1AU increases from 470km/s to 660km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' In contrast, α varies between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='4 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='2 as wind speed changes from 400km/s to 600km/s as estimated using HELIOS data (Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This differ- ence implies that other mechanisms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' electron heat flux, omitted in our model play an important role in lo- cal electron heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Second, the density only changes slightly among runs with different wave amplitudes (Fig- ure 2) and thus density does not vary with the wind speed significantly (top row of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' However, it is well known that faster solar wind is generally less dense than slower solar wind and the mass flux only moder- ately depends on the solar wind speed (Wang 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Both the inner boundary plasma density and the ex- pansion factor can heavily affect the solar wind density, but they are both constant in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' A thor- ough parametric study in which all these parameters are treated as variables is necessary and will be conducted in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Hence, the current study serves as a demon- stration of how the Alfv´en waves can contribute to the observed T−V correlations in the solar wind, while other mechanisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Fisk 2003) may still be important and should be incorporated in a complete description of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' This work is supported by NASA HTMS 80NSSC20K1275 and the NASA Parker Solar Probe Observatory Scientist grant NNX15AF34G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' The instru- ments of PSP were designed and developed under NASA contract NNN06AA01C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Kun Zhang for many useful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQf8PpF/content/2301.00852v1.pdf'} +page_content=' Software: Matplotlib (Hunter 2007) REFERENCES Adhikari, L.' metadata={'source': 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+Guangzhou, China +eraserhuang@tencent.com +Zhenyu Lei +Tencent Inc., +Guangzhou, China +rainylei@tencent.com +Yuetang Deng +Tencent Inc., +Guangzhou, China +yuetangdeng@tencent.com +Abstract—Code completion has become a common practice +for programmers during their daily programming activities. It +aims at automatically predicting the next tokens or lines that +the programmers tend to use. A good code completion tool +can substantially save keystrokes and improve the programming +efficiency for programmers. Recently, various techniques for code +completion have been proposed for usage in practice. However, it +is still unclear what are practitioners’ expectations on code com- +pletion and whether existing research has met their demands. To +fill the gap, we perform an empirical study by first interviewing +15 practitioners and then surveying 599 practitioners from 18 +IT companies about their expectations on code completion. We +then compare the practitioners’ demands with current research +via conducting a literature review of papers on code completion +published in premier publication venues from 2012 to 2022. Based +on the comparison, we highlight the directions desirable for +researchers to invest efforts towards developing code completion +techniques for meeting practitioners’ expectations. +Index Terms—code completion, empirical study, practitioners’ +expectations +I. INTRODUCTION +Code completion aims to predict the code tokens or lines +that programmers tend to input in their daily programming ac- +tivities. It can substantially save keystrokes and improve cod- +ing efficiency for programmers. As reported by [1], with code +completion, programmers can averagely save 3.65 keystrokes +for completing a token. Code completion has become the most +frequently-used feature of modern integrated development en- +vironments (IDEs) [2], and arouse increasingly more attention +from both academia and industry [3]–[13]. +Traditional code completion techniques [14]–[16] adopt +static analysis and provide completion candidates according to +pre-defined rules, which generally require non-trivial manual +efforts and perform ineffectively. To automate the whole +process, machine learning (ML)-based methods [17]–[19] such +as N-gram models are proposed to learn the code statistics, and +thereby can generalize to new code. With the development of +deep learning (DL), many DL-based approaches [3], [20] have +been proposed to implicitly learn the patterns in code and +achieve state-of-the-art performance. Some of the DL-based +techniques [5], [8], [10], [21] employ pre-trained language +(PLMs) such as BERT and GPT [22]–[24] for code com- +pletion, since PLMs are trained on large unlabelled corpora +and can encode amount of code knowledge into large-scale +parameters. Besides accurately predicting the next tokens, +PLM-based methods can also perform well in predicting the +next lines of code. +Despite numerous research efforts on code completion, +unfortunately, no prior studies have investigated practitioners’ +expectations on the techniques. It is unclear whether practi- +tioners appreciate the current code completion techniques and +what aspects (e.g., completion efficiency and effectiveness) +they care most about during adoption. The practitioners’ +perspective is important to unveil critical problems and can +provide guidance for software engineer researchers to create +solutions that satisfy programmers. +In this paper, we follow a mixed-methods approach [25], +[26] to gain insights into practitioners’ expectations on code +completion. We start with semi-structured interviews with 15 +professionals who have an average of 8.4 years of software +programming experience. Through the interviews, we qualita- +tively investigate the state of code completion practices, issues +faced by our interviewees when using code completion tools, +and their expectations on code completion. We then perform +an exploratory survey with 599 professionals from 18 IT +companies to quantitatively validate practitioners’ expectations +uncovered in our interviews. We finally conduct a literature +review of research papers published in premier venues from +2012 to 2022 and compare the techniques proposed in the +papers against the expectations of practitioners. +Specifically, we investigate the following four research +questions: +RQ1: What is the state of code completion practices? +This research question studies code completion practices +, including code completion tools and usage scenarios (the +scenarios practitioners use code completion). For completion +tools, 72% and 54% participants express that they often use +code completion in daily programming and are eager for a +better code completion tool, respectively. Meanwhile, built- +in tools in IDEs are far more popular than third-party plug-in +completion tools such as Copilot [11] and IntelliCode [8], used +arXiv:2301.03846v1 [cs.SE] 10 Jan 2023 + +by 96% and about 13% of the participants, respectively. +For usage scenarios, to facilitate the investigation, we +broadly divide code completion techniques according to the +completion granularities, including token level and statement +level. On average, 81% participants adopt token-level comple- +tion, which is evidently more than those adopting statement- +level completion (only 32%). For token-level completion, +84% and 85% participants often utilize tools to complete +identifiers and recommend APIs, respectively. For statement- +level completion, completing the currently edited line and +predicting the API argument are the two most popular usage +scenarios. +RQ2: Is code completion important for practitioners, and +what are the issues? +In this research question, we investigate how practitioners +perceive the importance of code completion, and study the +issues faced by practitioners during programming. For the im- +portance perceived by the practitioners, 87% of them strongly +agree or agree with the importance of code completion during +software development. For the issues of code completion, only +36% of the participants think current code completion tools +are satisfying. 56% and 58% participants consider erroneous +completion and painful tool installation as the main issues, +respectively. +RQ3: What are practitioners’ expectations on code com- +pletion tools? +This research question focuses on practitioners’ expectations +on different granularity level code completion tools, including +token-level and statement-level completion. We investigate the +expectations from multiple aspects including usage scenar- +ios, evaluation metrics, access to service (online or offline), +completion effectiveness, and efficiency. On average, 88% and +57% participants expect token-level completion scenarios and +statement-level completion scenarios, respectively. For token- +level completion, about 90% participants expect tools to com- +plete identifiers, recommend APIs, and complete paths (i.e., +when the required tokens are related to a file/directory path). +79% of participants are satisfied if the completion tools pro- +vide appropriate tokens within the top three candidates. Most +participants (85%) expect tools to generate completion results +within 200 milliseconds. For statement-level completion, the +most expected scenarios are skeleton completion (predicting +the skeleton of classes and methods), API argument rec- +ommendation (recommending the arguments of called APIs) +and completion of currently edited line. The most favored +completion latency is no more than 2 seconds. For evaluation +metrics, most participants care about the completion accuracy +(79%) and grammatical correctness (86%) for adopting token- +level and statement-level completion techniques, respectively. +RQ4: How well current research satisfies practitioners’ +demands? +This research question investigates the code completion +research and explores the gap between the research and +practitioners’ expectations. We have identified 26 papers about +code completion techniques, among which 17 papers propose +token-level completion and 10 papers are about statement- +level completion1. For the research about token-level com- +pletion, most of them focus on API recommendation and +identifier completion scenarios, which is consistent with prac- +titioners’ expectations. However, no papers have covered the +path completion scenario, a scenario expected by the large +majority of practitioners. For the research about statement- +level completion, most papers focus on next line completion, +but only a few papers explore skeleton completion and API +argument prediction that practitioners expect most. Besides, +most papers measure overlapped n-grams and edit similarity +between completed code and human-written code, however, +they are not preferred by the majority of the participants. +Our research aims at providing the future research direction +on code completion and helping researchers to consider the +expectations of practitioners when studying code completion +techniques. We hope researchers to propose better code com- +pletion tools, which can be eventually adapted into program- +ming practices and satisfy a wide range of programmers. +In summary, we make the following contributions: +• We are the first to shed light on the practitioners’ expec- +tations on code completion tools. We first interview 15 +and then survey 599 practitioners from 18 IT companies. +We investigate their opinions on code completion tools, +the scenarios they use for code completion, and their +expectations on code completion tools. +• We comprehensively review the papers published in the +premier venues in software engineering and artificial +intelligence fields in the last ten years. Then we compare +the published papers with the practitioners’ expectations, +highlighting the aspects desirable for researchers to be +improved in code completion for meeting the demands +of practitioners. +Paper Structure: Section II describes the methodology of +our study. Section III presents the results of our study. We +discuss the implications of our results in Section IV. Section +V discusses related work. +II. RESEARCH METHODOLOGY +In this section, we introduce our overall research method- +ology. Our research consists of three main stages. Stage 1: +Interviewing with professionals about their practices on code +completion, the issues they have met, and their expectations +on code completion tools. Stage 2: Performing an online +survey to validate and extend practitioners’ expectations based +on the interview. Stage 3: Conducting a literature review to +analyze whether and to what extent current techniques have +satisfied practitioners’ demands. The interviews and survey +were approved by the relevant institutional review board (IRB). +A. Stage 1: Interview +1) Protocol: Two authors conduct a series of in-person, +semi-structured, and in-depth interviews based on an inter- +view guide to explore the participants’ practices, issues, and +1The method proposed in [10] can conduct both token-level and statement- +level completion. + +Fig. 1: Example for illustrating the API argument prediction +scenario in the survey. +expectations on code completion. We invite 15 programming +professionals from seven IT companies worldwide to partici- +pate in the interviews. Each interview lasts 45-60 minutes. +In each interview, we first ask the interviewees some demo- +graphic questions about their background including job roles +and working experience. Then, we ask the interviewees to +freely talk about what they regard as a good code completion +tool and their expectations of code completion techniques. In +the end, we investigate their code completion practices and +the issues they have ever met. +2) Interviewees: We invite interviewees from our networks +in the IT industry. The interviewees are working full-time in +different roles such as developers and AI algorithms designers. +Eventually, 15 interviewees from seven IT companies agree to +participate in the interview. The programming experience of +our interviewees varies from 5 years to 13 years, with the +average programming experience at 8.4 years. +3) Data analysis: The first author transcribes the interviews +and summarizes the interviewees’ perspectives into opinion +cards. Then another author verifies the summarized opinion +cards and provides suggestions for improvement. After incor- +porating the suggestions, the two authors separately analyze +and sort the opinion cards into potential descriptions for the +questionnaire. The Cohen’s Kappa value between the two +authors is 0.67, indicating a substantial agreement between +them. The two authors discuss their disagreements to reach a +common decision. To reduce the bias of the two authors in +sorting descriptions, another two authors have also reviewed +and confirmed the final set of survey descriptions. +At the end of the interview, we derive four code completion +practices and six issues. We summarize three and six usage +scenarios for practitioners’ expectations on token-level and +statement-level code completion, respectively. Furthermore, +we derive eight and thirteen factors which specifically affect +the adoption of token-level and statement-level completion, +respectively, and draw another three aspects of general code +completion tools. +B. Stage 2: Online Survey +1) Survey Design: The survey consists of different types of +questions including single/multiple-choice, Likert scale (Op- +tions: Strongly Disagree, Disagree, Neutral, Agree, Strongly +Agree), and short answer questions. For each question, we +add an “I don’t understand” choice to filter the cases that +participants do not understand our descriptions. +The survey consists of six parts: +• Demographics. The survey first asks for demographic +information of the participants, including primary job +roles and programming experience. +• Code completion tools. We ask the participants about +the factors about code completion tools. In particular, we +investigate the code completion tools they have ever used +and their practices on using code completion tools. +• Code completion usage scenarios. This part aims at in- +vestigating in what scenarios practitioners use code com- +pletion tools. We summarize three scenarios for token- +level completion and six scenarios for statement-level +completion. For each scenario, we utilize a screenshot or +GIF for illustration, with an example shown in Figure +1. The three token-level completion scenarios include +identifier completion, API recommendation, and path +completion (i.e., completing the path of a referred file or +directory). The studied statement-level scenarios include +next line prediction, completion of currently edited line, +API argument recommendation, string completion (e.g., +completing a log), skeleton prediction and block content +prediction (e.g., predicting a method block). +• Tool importance. We investigate the practitioners’ at- +titudes towards the code completion tools in this part. +We ask the participants about how they perceive the +importance of the tools. +• Code completion issues. This part investigates the issues +faced by the participants during using the code comple- +tion tools, such as erroneous completion, high completion +latency, high resource consumption and concern about +data leak. +• Practitioners’ expectations. In this part, we study prac- +titioners’ expectations on code completion tools of the +two granularity level, i.e., token-level completion and +statement-level completion. For each granularity level, we +ask multiple aspects of the expectations including usage +scenarios, evaluation metrics, access to service (online or +offline), completion effectiveness and efficiency. Besides, +we also investigate general aspects that affect practition- +ers’ likelihood to adopt code completion tools such as +time consumption of tool installation . +At the end of the survey, we allow our participants to choose +to freely provide their comments, advice, and opinions about +code completion and our survey. +To check participants’ perceptions about the survey length +and clarity of the descriptions, we conduct a preliminary +survey with a small set of practitioners that are different +from our interviewees and survey participants before large- +scale distribution. Based on the received feedback, we make +minor modifications to the survey and produce a final version. +We utilize widely-used questionnaire websites [27], [28] to +distribute the survey. +2) Participant Recruitment: We contact professionals work- +ing full time in IT companies in our social networks, and +ask for their help to complete and disperse the survey. In +particular, we send invitations to the professionals working in +Microsoft, Intel, Tencent, Alibaba, and other IT companies. + +1 +from sklearn.ensemble import RandomForestclassifier +2 +X_train = [1,2,3,4] +3 +Y_train = [1,0,1,0] +4 +model = RandomForestclassifier(n estimators=2) +5 +model.fit(X_train, Y_train)TABLE I: Participants roles & working experience +Role +Population +<1y +1-3y +3-5y +5-10y +>10y +Development +345 +32 +102 +121 +65 +15 +Algorithm Design +173 +10 +46 +81 +42 +4 +Testing +27 +6 +8 +4 +6 +3 +Architect +4 +0 +1 +2 +1 +0 +Project Manager +2 +0 +0 +0 +1 +1 +Others +48 +9 +15 +17 +7 +0 +599 +57 +172 +225 +122 +23 +We finally receive 611 survey responses in total with the +average completion time at 8.2 minutes. Among them, we dis- +card 12 responses that are completed within two minutes. The +survey results presented in this paper are analyzed from the +remaining 599 valid responses. An overview of the surveyed +participants’ roles and their experience is depicted in Table I. +Most participants are engaged in software development and +have 3-5 years of programming experience. +3) Result Analysis: We analyze the results based on the +question types. For multiple-choice and single-choice ques- +tions, we report the percentage of each selected option. For +Likert-scale questions, we draw bar charts to illustrate the +distributions of the Likert scores. For the open-ended short +answer questions, we conduct a qualitative analysis of the +results. Besides, we drop “I don’t understand” ratings that form +a small minority (less than 1%) of all the received ratings. +C. Stage 3: Literature Review +The papers about code completion are usually published in +software engineering and artificial intelligence fields. There- +fore, we go through research papers published in ICSE, +ESEC/FSE, ASE, ICPC, SANER, MSR, ICSME, PLDI, OOS- +PLA, TSE, TOSEM, EMSE, ACL, EMNLP, NAACL, IJCAI, +ICLR, NeurIPS, and AAAI from 2012 to 2022. We select +papers from the above conferences and journals because they +are premier publication venues in software engineering and +artificial intelligence fields. +We first read the title and abstract of the papers to check +whether they are related to code completion. For each code +completion paper, two authors read its content and analyze the +capabilities of the proposed approach in terms of completion +granularity, scenarios, evaluation metrics, and the access to +the completion service, and efficiency. For instance, Izadi et +al. [10] declare that they utilize multi-task learning to train +the model to predict next token and next statement. Thus, we +infer that this paper works on both token-level and statement- +level completion. Svyatkovskiy et al. [3] claim to provide web +completion services and client completion model, and we infer +the access to its service to be both online and offline. Two +authors discuss the differences in the capability analysis and +confirm the final results through further paper reading. +III. RESULT ANALYSIS +A. RQ1: Code completion practices +In this research question, we investigate the practitioners’ +code completion practices, including the used code completion +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Others +AiXcoder +Kite +Tabnine +Copilot +IntelliCode +Compiler-based tool +IDE built-in tool +(a) Statistics of code completion tools that practitioners have used. +0% +10% +20% +30% +10% 20% 30% 40% 50% 60% 70% 80% +Percentage of Valid Responses +I am eager for a better +code completion tool [P3] +I find programming laborious +without code completion tools [P2] +I often use code completion tools [P1] +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +(b) Usage status of code completion tools. +Fig. 2: Code completion tools. +tools and usage scenarios. The participants’ rating results for +some descriptions related to code completion practices are +illustrated in Figure 2 and 3. +1) Code completion tools: We first ask the participants +what code completion tools they have used and the results +are shown in Figure 2 (a). From the results, we find that 96% +participants express they adopt IDE built-in code completion +tools. Compiler-based tools such as CCLS and Clangd [14] are +the second most popular tools among surveyed practitioners. +However, the third-party plug-in tools are not prevalent, e.g., +about 13% participants have used Copilot [11] and IntelliCode +[8]. +We then investigate the practitioners’ usage status of code +completion tools, with results illustrated in Figure 2 (b). Al- +though most practitioners (72%) indicate that they often adopt +code completion in daily programming, only 50% participants +agree that they find programming laborious without code +completion. Besides, 54% of them are proactive in seeking +a better code completion tool. “I always keep tuned on the +release of new code completion tools and am eager to be the +first to use them.”, as a participant stated. +Finding 1: The most commonly-used code completion tool +is IDE built-in tool. Other third-party plub-in tools such as +IntelliCode [7] and Copilot [11] are far less popular, for +which only about 13% participants have used. In addition, +most participants express that they often use code comple- +tion tools, and 54% of them are eager for a better code +completion tool. +2) Usage scenarios: In this section, we investigate in what +scenarios the practitioners use code completion. According +to [29], we study the scenarios from two granularities in- +cluding token-level and statement-level completion, respec- +tively. Token-level completion aims at completing the currently +typing token and the next token to be used for developers. +Statement-level completion is able to predict multiple tokens +and even lines of code. We provide the definitions of the +two granularity level completion in the survey for facilitating +participants’ understanding. The token-level completion and + +0% +10% +20% +30% +40% +50% +60% +70% +10% 20% 30% 40% 50% 60% 70% 80% 90% +Percentage of Valid Responses +Block content prediction [C9] +Skeleton prediction [C8] +String completion [C7] +API argument recommendation [C6] +Completion of currently edited line [C5] +Next line prediction [C4] +Path completion [C3] +API recommendation [C2] +Identifier completion [C1] +Token +Statement +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +Fig. 3: Scenarios that practitioners use code completion. +0% 10% 20% 30% 40% 50% 60% 70% 80% 90% +Percentage of Valid Responses +Code completion improves programming experience [M5] +Code completion can reduce my typing mistakes [M4] +Code completion can provide some hint [M3] +Code completion can save my key strokes [M2] +Code completion can improve my efficiency [M1] +I think code completion is important +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +Fig. 4: Importance of code completion. +statement-level completion involve three and six scenarios, +respectively, as introduced in Section II-B. The participants’ +ratings of the usage scenarios are shown in Figure 3. We ob- +serve that the overall popularity of statement-level completion +scenarios is obviously less than that of token-level (32% v.s. +81% on average). +Token-level completion. As can be seen in the upper part +of Figure 3, all the three usage scenarios are popular among +the practitioners. Specifically, about 85% participants use code +completion to complete identifiers and recommend APIs. More +than 70% survey practitioners adopt code completion for path +completion in their daily programming. +Finding 2: All the three token-level code completion usage +scenarios are popular among the participants, among which +identifier completion and API recommendation are the most +widely used scenarios. +Statement-level completion. As shown in the lower part of +Figure 3, the three most popular statement-level completion +scenarios are API argument recommendation, completion of +currently edited line, and skeleton prediction, used by 46%, +45%, and 36% participants, respectively. In addition, about +25% of them adopt code completion to predict the next line of +code and string content. Only 16% of survey participants adopt +code completion in the block content prediction scenario. One +participant said that “The performance of existing completion +tools is not good enough to predict the whole block content. +The predicted results may be severely misleading in some +cases, which reduces my programming efficiency.” +Finding 3: The most adopted statement-level code comple- +tion scenarios are API argument recommendation, comple- +tion of currently edited line, and skeleton prediction. +0% +10% +20% +30% +40% +50% +60% +10% 20% 30% 40% 50% +Percentage of Valid Responses +Painful tool installation [I6] +Erroneous completion [I5] +Concern about data leakage [I4] +High resource consumption [I3] +High completion latency [I2] +Unsatisfying ranking [I1] +The tools are satisfying for me +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +Fig. 5: Issues of code completion. +B. RQ2: Code completion Importance and Issues +1) Importance of Code Completion: We ask whether they +think code completion is important for development, and their +agreements on the possible reasons of the importance. The +results are illustrated in Figure 4. +From the results, we can find that most participants (88%) +think that code completion plays an important role. 90% +of them agree or strongly agree that code completion can +provide some hint on the implementing code. One practitioner +described that “Sometimes I forget how to type a long identifier +or method name. In this case, code completion can show +me the candidates and provide some hint for completing the +code.”. Besides, more than 85% of them think completion tools +are helpful to reduce typing mistakes, improve programming +efficiency, save keystrokes, and improve programming experi- +ence. +Finding 4: 88% of the survey participants agree that code +completion tools play an important role in programming. +Most participants (90%, 87% and 86%) think the tools can +provide some hint, improve programming efficiency, and +improve their programming experience. +2) Code completion issues: Figure 5 illustrates the partici- +pants’ ratings of code completion-related issues they faced dur- +ing programming. Despite the importance of code completion, +only 36% participants think the tools they are currently using +are satisfying. 45% and 44% participants consider erroneous +completion and painful tool installation as the main issues, +respectively. Erroneous completion indicates that some errors +occur in the completed code, which practitioners need to +modify. As a participant said: “The completion is usually not +exactly what I need, and it is annoying for me to rectify the +errors.”. Besides, 34% and 31% of them are not satisfied +with the rankings of appropriate tokens in the candidates +and concerned about data leakage in the tools, respectively. +Among the participants, about a quarter of them hold the view +that the high resource (i.e., memory and CPU) consumption +and high completion latency are not acceptable. A participant +working on large C++ projects told us: “The C++ code +completion tools like CCLS and Clangd parse the opened cpp +files and build indexes for them when programming, which +may consume all of my CPU and memory.”. +Finding 5: The most concerned issues by practitioners are +the erroneous code completion, painful tool installation, and +unsatisfactory ranking. + +0% +10% +10% +20% +30% +40% +50% +60% +70% +80% +90% +Path completion [T3] +API recommendation [T2] +Identifier completion [T1] +(a) Expectations on usage scenarios. +0% +10% +10% +20% +30% +40% +50% +60% +70% +80% +Recall [T6] +Average Rank [T5] +Accuracy [T4] +(b) Expectations on evaluation metrics. +0% +10% +20% +30% +10% +20% +30% +40% +50% +60% +70% +80% +Offline [T9] +Internal Network [T8] +Public Network [T7] +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +(c) Expectations on access to completion service. +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +100% +Rank of the needed token in +completion candidates [T10] +1 +2 +3 +4 +5+ +(d) Expectations on code completion effectiveness. +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +100% +Maximum completion latency [T11] +100ms +200ms +300ms +400ms +600ms +800ms +1000ms +(e) Expectations on code completion efficiency. +Fig. 6: Expectations on token-level code completion, where +horizontal axis denotes percentage of valid responses. +C. RQ3: Practitioners’ Expectation +In this research question, we comprehensively investigate +the practitioners’ expectations on code completion. We explore +the aspects including usage scenarios, evaluation metrics, +access to service, completion effectiveness, and efficiency. We +report practitioners’ expectations on the two levels of granu- +larity of code completion, i.e., token-level and statement-level +completion. We also investigate practitioners’ expectations on +three other aspects of general completion tools that impact +their usage experience, including time consumption of instal- +lation, personalized completion and additional information +display. +1) RQ 3.1 Expectations on token-level completion: The +participants’ ratings for the expectations on token-level code +completion is shown in Figure 6. +Usage scenarios. From Figure 6 (a), most participants show +positive attitudes towards the three scenarios. More than 88% +of them expect to use code completion in Identifier completion +and API recommendation scenarios. Besides, 86% of partici- +pants expect code completion tools to complete the path they +are typing. In one participant’s opinion, path completion is +very important when he works on a large project, because +he/she finds it very easy to make mistakes on file names +without path completion. +Finding 6: For token-level code completion, more than 80% +participants agree that tools are supposed to support identi- +fier completion, API recommendation and path completion. +Evaluation metrics. We study participants’ opinions on eval- +uating token-level code completion tools in Figure 6 (b). We +observe that Accuracy gains the most support rate (79%). +Besides, more than 70% of participants also care about Av- +erage rank of the needed token in the candidates (e.g., Mean +Reciprocal Rank MRR) and Recall of the tool. +Access to service. Figure 6 (c) shows that 82% of partici- +pants expect to use token-level completion tools offline. One +participant stated that: “Offline tools are the most stable ones, +which will not be affected by network status and I can use +them anywhere.”. In contrast, the support rate of online tools +is relatively low. 62% and 50% of participants are willing to +use the online tools in the internal network and in the public +network, respectively. +Effectiveness. Figure 6 (d) shows the practitioners’ satisfac- +tion ratio against different ranks of needed token in completion +candidates. From our survey, if the code completion tool can +rank the needed token in the top 3 candidates, it will satisfy +about 80% practitioners. +Efficiency. We show the participants seven GIFs with different +completion latency including 100, 200, 300, 400, 600, 800, and +1000 milliseconds. From Figure 6 (e), we find that if a token- +level code completion tool can give completion candidates +within 200 milliseconds, it can satisfy 85% practitioners. +2) RQ 3.2 Expectations on statement-level completion: +The results of participants’ expectations on statement-level +completion can be accessed in Figure 7. +Usage scenarios. The practitioners’ expectations on the +statement-level code completion scenarios are illustrated in +Figure 7 (a). We observe that practitioners’ expectations show +a similar trend with the popularity of usage scenarios as shown +in Section III-A. Likewise, The three most expected scenarios +are skeleton prediction, completion of currently edited line +and API argument recommendation, where all of them gain +at least 60% support rate, and skeleton prediction is the most +expected. Besides, 51% participants agree that statement-level +completion tools are supposed to predict block content such as +method block and loop block. 53% of them expect to use code +completion tools to predict strings. Interestingly, though the +scenario of predicting the next line of code is kind of similar to +completing the currently edited line, only 43% of participants +regard next line prediction as an important scenario while the +currently edited line completion obtaining a 62% support rate. +As one participant commented: “When finishing a line of code, +I will start to type the next line immediately instead of waiting +for completion results at the end of the line.”. +Finding 7: skeleton prediction, completion of currently +edited line and API argument recommendation are consid- +ered to be the most expected statement-level code comple- +tion scenarios, which are similar as the popularity of the +usage scenarios. + +0% +10% +20% +30% +40% +10% +20% +30% +40% +50% +60% +70% +80% +Block content prediction [S6] +Skeleton prediction [S5] +String completion [S4] +API argument recommendation [S3] +Completion of currently edited line [S2] +Next line prediction [S1] +(a) Expectations on usage scenarios. +0% +10% +20% +10% +20% +30% +40% +50% +60% +70% +80% +90% +Grammatical correctness [S11] +Readability [S10] +Function similarity [S9] +Edit similarity [S8] +Overlapped N-grams [S7] +(b) Expectations on evaluation metrics. +0% +10% +20% +30% +10% +20% +30% +40% +50% +60% +70% +Offline [S14] +Internal Network [S13] +Public Network [S12] +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +(c) Expectations on access to completion service. +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +100% +Minimum grammatical correctness rate [S16] +Maximum manual modification rate [S15] +20 +40 +60 +80 +100 +(d) Expectations on code completion effectiveness. +0% +10% +20% +30% +40% +50% +10% +20% +30% +40% +Which one is more important? +Effectiveness or Generated Code Quantity [S17] +Effectivess is much more important +Effectivess is more important +Neutral +Generated code quantity is more important +Generated code quantity is much more important +(e) Effectiveness v.s. Generated code quantity. +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +100% +Average time consumption of +completing a line of code [S18] +<0.5s +0.5~1s +1~1.5s +1.5~2s +>2s +(f) Expectations on average time consumption of generating a line of code. +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +100% +Maximum time consumption of +giving completion results [S19] +<0.5s +0.5~2s +2~5s +5~10s +>10s +(g) Expectations on maximum time consumption of giving completion +results. +Fig. 7: Expectations on statement-level code completion, +where the horizontal axis denotes percentage of valid re- +sponses. +Evaluation metrics. We illustrate the practitioners’ expecta- +tions on different evaluation metrics for statement-level code +completion approaches in Figure 7 (b). The top three preferred +evaluation metrics are grammatical correctness, +readability +and functional/structure similarity, supported by more than +80% participants. However, the overlapping n-grams (e.g., +BLEU score [30]) and edit similarity receive the least support +rate (i.e., 65% and 70%, respectively). +Access to service. Figure 7 (c) presents the likelihood of dif- +ferent access to use a statement-level code completion tool. Al- +though the support rate of offline service drops 15% compared +to token-level completion (from 82% to 67%), offline tools +are still the most expected. However, the willingness to adopt +an online statement-level code completion tool increases. For +instance, 66% participants agree or strongly agree to invoke +online completion services in internal networks (compared +with 62% for token-level completion). Furthermore, the sup- +port rate of public-net tools also increases from 50% to +58%. One participant told us: “Currently almost all statement- +level completion tools are AI-based, which means that their +computation complexity is high. Thus, utilizing online code +completion service is acceptable.”. Moreover, we observe +the difference between the participants’ expectations on tools +deployed in public and internal network (58% and 66%). +According to our survey, the difference can be attributed to +the concern about data leakage and network stability in public +network. As a participant stated that in his/her department, +Copilot was not permitted to be installed because it would +upload their code to the cloud and might result in leakage of +confidential code. +Effectiveness. From Figure 7 (d), we find that only 26% +participants can accept the ratio of manual modification greater +than 60% of completed code. Besides, 35% and 39% of our +participants expect the maximum modification ratio to be no +more than 20% and 40%. For grammatical correctness, if +a statement-level code completion tool ensures that at least +80% of the completed code is grammatically correct, it can +satisfy 82% practitioners. Besides, 18% participants have strict +requirements on the correctness of generated code, i.e., they +expect the tools to make no mistake on code grammar. +According to existing work [9], [21], the more code is com- +pleted, the more errors occur. We further investigate the practi- +tioners’ preference between completion effectiveness (generate +less code with higher accuracy) and generated code quantity +(generate more code with more manual modifications). The +results are illustrated in Figure 7 (e). From our survey, there +is no very clear winner between them (46% support rate for +effectiveness and 40% for generated code quantity), and a few +more participants consider effectiveness more important. In +this question, participants mainly have two points of view: +1) Effectiveness is more important. For the supporters of +effectiveness, completion results with errors may affect +their efficiency. A participant shared his/her experience +with statement-level code completion tools. “It was very +annoying for me to modify the completion results. ... I’d +rather have less code completed than modify completion +results.” +2) Code quantity is more important. Supporters of code +quantity expect statement-level completion tools to pre- +dict more code at once, even if the code needs to be +modified. One participant stated that as long as the tool +could predict the code with the right logic (e.g., correct +code skeleton), the tool was effective in his eyes. +Efficiency. We study practitioners’ expectations on the effi- +ciency of statement-level code completion tools from two as- +pects including average time consumption of completing a line +(shown in Figure 7 (f)) and maximum latency of generating +completion results (shown in Figure 7 (g)), respectively. +From our survey, we observe that the speed of averagely +taking 0.5 seconds to generate a line receives the highest + +0% +10% +20% +30% +40% +50% +60% +70% +80% +90% +Maximum Installation + time consumption [O1] +1min +5mins +10mins +20mins +Other +(a) Expectations on time consumption of tool installation. +0% +10% +10% +20% +30% +40% +50% +60% +70% +80% +90% +I hope code completion tools can +learn my programming habits [O3] +I hope code completion tools can +provide additional information [O2] +Strongly Disagree +Disagree +Neutral +Agree +Strongly Agree +(b) Expectations on additional information display and personalized com- +pletion. +Fig. 8: Expectations on other aspects of general code com- +pletion tools, where the horizontal axis denotes percentage of +valid responses. +support rate at 44%, and 39% participants expect the time +to be no more than 1 second. For the maximum completion +latency, the practitioners who expect the code completion tools +to generate completion results in less than 2 seconds have +the largest proportion (i.e., 47%). If a statement-level code +completion tool generates completion results within 2 seconds, +it can satisfy 67% practitioners. In addition, few participants +(i.e., 4%) express they are willing to wait for more than 5 +seconds for completion. +Finding 8: For completion efficiency, most practitioners +expect statement-level code completion tools to take less +than 0.5 seconds to complete a line of code on average. Be- +sides, to meet the majority (67%) of practitioners’ demands, +tools are supposed to generate completion results within 2 +seconds. +3) RQ 3.3 Expectations on other aspects of general code +completion tools: In this section we present the practitioners’ +expectations on general aspects of code completion, and the +results are illustrated in Figure 8. +Figure 8 (a) shows that practitioners’ expectations on the +time cost of installing code completion tools. 83% participants +will be satisfied if they can install the tool within 5 minutes. +Besides, we observe that 14% of participants are willing +to spend more than 20 minutes on installing tools. As one +participant said: “I would like to spend a whole day on +installing a code completion tool as long as it can really +improve my efficiency.” +From Figure 8 (b) we observe that most (76%) practition- +ers agree or strongly agree that code completion tools are +supposed to learn their programming habits (e.g., preference +for function names and variables). In addition, 83% of them +expect code completion tools to simultaneously present some +additional information about the predicted code (e.g., showing +users the API definition or documentation while recommend- +ing an API). One participant told us: “Sometimes I forgot the +order of arguments in the API, thus showing me how the API +is defined can greatly reduce my programming mistake.” +TABLE II: Capabilities of current research. Likert score de- +notes the average weighted score of participants’ ratings (1 to +5 correspond to strongly disagree to strongly agree). +Description +Likert +Papers +Score +Token-level completion +Usage Scenarios +Identifier completion +[T1] +4.50 +[5], [10], [31]–[35] +API recommendation +[T2] +[3], [17], [31], [36]–[38] +4.50 +[4]–[6], [32], [33], [35] +[7], [10], [19], [34], [39] +Path completion +[T3] +4.39 +- +Evaluation Metrics +Accuracy +[T4] +4.19 +[10], [17], [32], [34] +Average rank +[T5] +4.00 +[3], [5], [7], [36], [38] +Recall +[T6] +[19], [33], [34], [36] +4.05 +[7], [31], [35], [39] +[3], [6] +Others +[17], [32], [36], [37] +Access to service +Online +[T7] +3.43 +[3], [10] +Offline +[T9] +4.20 +[3], [7], [17] +Time Consumption +Mentioned +[6], [32]–[34], [38], [39] +[3], [7], [10], [17], [36] +Statement-level completion +Usage Scenarios +Next line prediction +[S1] +3.27 +[8]–[10], [21] +[20], [37], [40], [41] +Completion of currently edited line +[S2] +3.74 +[8], [20], [21] +API argument recommendation +[S3] +3.81 +[8], [21], [34] +String completion +[S4] +3.32 +- +Skeleton prediction +[S5] +3.88 +- +Block content prediction +[S6] +3.43 +[21], [40], [42] +Evaluation Metrics +Overlapped n-grams +[S7] +3.72 +[9], [21], [37], [40] +Edit similarity +[S8] +3.81 +[8], [37], [40], [42] +Function similarity +[S9] +4.09 +[9] +Readability +[S10] +4.31 +- +Grammatical correctness +[S11] +4.42 +- +Others +[8], [20], [40]–[42] +Access to service +Online +[S12] +3.54 +[8], [10], [42] +Offline +[S14] +3.89 +[41] +Time Consumption +Mentioned +[8], [20], [40], [41] +D. RQ4: Current State-of-the-art Research +After our literary review, we identify 26 papers in total +from the top conference and journals in software engineering +and artificial intelligence communities. Table II shows the +capabilities of surveyed code completion techniques. We can +observe that the research on token-level completion is much +more than that on statement-level completion (17 papers v.s. +10 papers). +1) Token-level completion: Usage scenarios. As seen from +Table II, all of the 17 collected token-level completion papers +support API recommendation. However, only a few of these +papers work on identifier completion, and none of them +mention path completion. This may be attributed to that most +identifiers and path tokens are Out-of-Vocabulary (OoV) words +for the proposed techniques. +Evaluation metrics. In this part, accuracy is equivalent to top- +1 accuracy and error rate, recall refers to the more general top- +k accuracy, and average rank refers to the average rankings of +the appropriate token in the prediction candidates (e.g., Mean +Reciprocal Rank, MRR). In Table II, we find that most of +the papers utilize recall and averaged rank to evaluate the +effectiveness of their approaches. Moreover, other evaluation +metrics are explored [3], [7], [17], [36]. Some researchers +attempt to explore better metrics as the proxy for users’ +productivity such as saved keystrokes [17]. +Access to service. Few papers have focused on employing + +their techniques in industrial products. We classify the papers +into two categories including online and offline according to +their deployment. Svyatkovskiy et al. [3], [7] develop their +system as part of Intellicode extension in Visual Studio Code +IDE [43], allowing programmers to use it offline. Izadi et +al. [10] utilize a Transformer architecture, claiming that they +put the model on the cloud and provide completion web +services. Moreover, Hindle et al. [17] mention that their tool +is incorporated into the offline Eclipse plug-in. +Efficiency. Among collected token-level code completion pa- +pers, 11 of them explicitly consider the time consumption. +Nine approaches proposed in [3], [6], [7], [10], [17], [32], [33], +[36], [38] take less than 200 milliseconds to predict a token. +According to our results, these techniques with such latency +can satisfy 85% practitioners. However, another two methods +take several seconds to complete a single token [34], [39], +which is considered unacceptable for token-level completion. +2) Statement-level completion: Usage scenarios. As seen +from Table II, most papers focus on next line prediction +which receives only 43% support rate, but the more expected +scenarios completion of currently edited line and API argument +recommendation obtain less attention. Three papers propose +approaches for block content prediction scenario. Moreover, +no papers propose techniques to predict the methods’ skele- +tons (skeleton prediction) and complete string content (string +completion). +Finding 9: Most papers focus on next line prediction +scenario, while only a few of them explore completion of +currently edited line and API argument recommendation. +Besides, no paper has proposed techniques for skeleton pre- +diction, which is considered as the most expected scenario. +Evaluation metrics. From Table II, most papers evaluate the +generated code via overlapped n-grams such as BLUE and +ROUGE, and edit similarity, which receive the least support +rate (i.e., 65% and 70%, respectively) according to our survey. +Only one paper evaluates the generated code against human- +written code via function/structure similarity. Besides, none +of them mention the readability and grammatical correctness +of the generated code. Some papers also propose customized +metrics to comprehensively evaluate their statement-level com- +pletion approaches such as click-through rate [8], [20], [40]– +[42]. +Finding 10: Most papers focus on measuring overlapped +n-grams and edit similarity between the generated code and +the human-written code that are not preferred by the large +majority of participants. No paper evaluates the generated +code via grammatical correctness and readability, which the +practitioners value most. +Access to service. The work [8] proposes a tool called +Intellicode Compose, which deploys the models on the cloud +and also allow client-side caching. Moreover, Wen et al. [42] +present an android studio plugin as a web service. In addition, +the tool proposed in [41] serves as an Eclipse plug-in and +provide offline completion service. +Efficiency. +Among +the +collected +papers +that +propose +statement-level code completion approaches, only four of them +explicitly discuss the time consumption of their techniques. +For example, the methods proposed in [8], [21], [41] take less +than 1 second to complete code statements, while the work in +[20] takes more than 5.5 seconds on average for prediction. +Finding 11: Time consumption, a critical adoption factor +of code completion tools, is missing in 40% of our collected +token-level and statement-level papers. +IV. DISCUSSION +A. Implications +Our survey results highlight several implications for the +research community: +1) Implication on code completion scenarios: For token- +level code completion tools, besides identifier completion and +API recommendation, they are also expected to support path +completion which is expected by 86% participants. +For the code completion tools that are capable of predicting +lines of code, the most important scenarios for programmers +are skeleton prediction, completion of currently edited line and +API argument recommendation. However, most current papers +about statement-level completion focus on less anticipated +scenarios such as next line prediction [9], [10]. +2) Implication on code completion tools: The large ma- +jority of programmers expect code completion tools to be +more intelligent. For instance, 76% participants expect the +tools to learn their programming habits so that they can keep +the programming style consistent and reduce modification. +Besides, most of them are also eager to be informed with +additional information about the predicted code such as API +definition or documentation (81%). One participant stated +that “It is hard for me to remember countless APIs in my +project. When I use an API, if the code completion tool +locates and shows the API documentation simultaneously, it +can significantly save time in searching the API.” +Besides, most practitioners wish the time consumption for +tool installation and configuration to be within 20 minutes. If +they have to spend much time on tool installation, they may +be dissuaded from using it. +3) Implication on evaluation metric: Evaluation metric +is another important factor that should satisfy practitioners’ +expectations. Most of existing studies about statement-level +code completion evaluate the generated code by comparing +with human-written code in terms of overlapping n-grams +(such as BLEU score and ROUGE) [21] and editing sim- +ilarity [29], [40]. However, these two metrics receive the +least support rate among all the evaluation metrics in our +survey. Practitioners more expect the tools to use the metric +function/structure similarity for evaluation. A participant told +us: “ ... BLEU score may be a good criteria to judge the quality +of generated natural language texts, however, it can hardly +evaluate whether a code snippet is satisfactory.” In addition, +the metrics that participants value most such as grammatical + +correctness and readability of generated code are missing in +current publications. +4) Balancing effectiveness and code quantity: From our +survey, 46% practitioners regard the effectiveness of statement- +level code completion weighs more than the quantity of pre- +dicted code (39%). Considering that two completion strategies +are both preferred by a certain number of participants, it will +be better if the tools provide a configurable option for users +to decide the quantity of predicted code. +Besides, we also observe that most practitioners expect +that the completed code does not need extensive manual +modification. For instance, 74% participants cannot accept that +more than 40% of completed code needs manual modification. +Thus, code completion tools may preliminarily estimate the +probability of whether over 40% of predicted code needs +modifying. If the probability is high, tools can terminate +current completion process and generate completion results. +5) Implication on code completion latency: Completion la- +tency is one key aspect that substantially affects practitioners’ +likelihood to adopt code completion tools. From our survey, +only 15% participants can accept that tools take more than +400 milliseconds to predict a token. Besides, 83% of them +wish the average time consumption of generating a line of +code to be less than 1 second, and 80% participants expect +the statement-level completion latency to be no more than +2 seconds. However, the time consumption factor is missing +in nearly half of our collected papers. Researchers should +pay more attention on code completion latency. Considering +that offline tools are the most expected, how to effectively +compress the completion models is an important direction for +future studies. +6) Improving robustness of code completion tools: In our +survey, many participants mention that the robustness of code +completion tools is also vital to user experience. Robustness +requires that the completion results are not affected by slight +perturbations in the input. From our survey, a participant +shared his/her experience with Tabnine: “Tabnine can suc- +cessfully predict the API arguments if the variable is named +‘X train’. However, if I modify the name to ‘x train’, the +completion results will be totally different.” Besides identifier +changes, code completion tools are also supposed to be robust +to statement changes. A participant stated that: “AiXCoder can +predict the block content of a method well, but the results +turn to be terrible when I inserted an unrelated assignment +statement.” +B. Threats to Validity +One of the threat in our survey is that there may be some +participants who do not fully understand the questions. For +instance, some participants have never used statement-level +code completion tools. Therefore, they may be unfamiliar with +the questions of statement-level code completion. To reduce +this threat, we utilize one or two clear images or GIFs to +describe each scenario and facilitate them better understanding +the questions. Furthermore, some participants do not answer +the questions seriously and the results cannot reflect their +beliefs. Therefore, we drop the responses completed less than +two minutes. This is a common and tolerable threat to validity +in previous studies, e.g., [44]. +Another threat is that our participants may not be represen- +tative in typical programmers. Our solution is to widely survey +practitioners working in many IT companies. We believe we +have made this threat have minimal impact on the results of +our survey. +V. RELATED WORK +A. Code Completion +Traditional code completion focused on static analysis tech- +niques associated with manually defined rules to suggest code +[15], [16], [45], [46]. For instance, researchers utilized type +information [16], [47], similar code snippets [48] and history +data [49] to predict needed code tokens. +Equipped with machine learning, a series of code com- +pletion work equipped with statistical language models was +proposed [17]–[19], [34], [36], [38]. For instance, Hellen et al. +[36] explicitly took the techniques such as nested scopes into +account and improved the performance of the n-gram model. +Moreover, Raychev et al. combined decision trees and domain +knowledge, proposing a probabilistic model [33]. +With the development of deep learning, neural networks +such as RNN [50] and Transformer [51] showed great ca- +pability to learn features from source code [5], [31], [36]. +For instance, Li et al. [31] proposed a point mixture network +to relieve the Out-of-Vocabulary problem. Kim et al. [5] +parsed the code into abstract syntax trees and fed the trees +into a Transformer-based model. In recent years, pre-trained +language models have been leveraged for predicting multiple +code tokens [8], [10]. Svyatkovskiy et al. [8] proposed GPT-C, +which could predict code statements. Izadi et al. [10] raised +CodeFill, which combined type and semantic information of +code, further improving the completion performance. +B. Studies on Code Completion Practices +Apart from research on code completion, some other work +focus on studying code completion practices [52]–[55]. For +instance, Vaithilingam et al. [52] asked participants to fin- +ish programming tasks with or without Copilot [11] and +determined whether Copilot is useful. In addition, Ziegler +et al. [53] focused on investigating evaluation metrics of +code completion. Authors compared the measurable user data +(objective data) and the user-reported productivity (subjective +data), and identified the most representative metric as a proxy +of productivity. The work [54], [55] both pointed out the differ- +ence between synthetic data and real-world data, and identified +the most important tokens that needed to be completed. +However, no prior studies have investigated the practi- +tioners’ expectations on code completion. In this paper, we +conduct a large scale user study, investigating practitioners’ +expectations on multiple aspects of code completion tools. +Moreover, we perform a comprehensive literature review to +reveal the gap between current techniques and practitioners’ +expectations, providing future directions for researchers. + +VI. CONCLUSION AND FUTURE WORK +In this paper, we interview 15 professionals and survey +599 practitioners on completion practices, issues they face +and their expectations on code completion tools. 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Li, “Learning autocompletion from real-world +datasets,” in 43rd IEEE/ACM International Conference on Software +Engineering: Software Engineering in Practice, ICSE (SEIP) 2021, +Madrid, Spain, May 25-28, 2021. +IEEE, 2021, pp. 131–139. + diff --git a/VNE2T4oBgHgl3EQfXgf7/content/tmp_files/load_file.txt b/VNE2T4oBgHgl3EQfXgf7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8632cf4e0c6892e7d233a2ace97ac94baf9f9381 --- /dev/null +++ b/VNE2T4oBgHgl3EQfXgf7/content/tmp_files/load_file.txt @@ -0,0 +1,1099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf,len=1098 +page_content='Practitioners’ Expectations on Code Completion Chaozheng Wang Harbin Institute of Technology, Shenzhen, China wangchaozheng@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='cn Junhao Hu Peking University, Beijing, China junhaohu@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='cn Cuiyun Gao Harbin Institute of Technology, Shenzhen, China gaocuiyun@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='cn Yu Jin Tencent Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', Guangzhou, China lenajin@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='com Tao Xie Peking University, Beijing, China taoxie@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='cn Hailiang Huang Tencent Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', Guangzhou, China eraserhuang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='com Zhenyu Lei Tencent Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', Guangzhou, China rainylei@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='com Yuetang Deng Tencent Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', Guangzhou, China yuetangdeng@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='com Abstract—Code completion has become a common practice for programmers during their daily programming activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' It aims at automatically predicting the next tokens or lines that the programmers tend to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' A good code completion tool can substantially save keystrokes and improve the programming efficiency for programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Recently, various techniques for code completion have been proposed for usage in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, it is still unclear what are practitioners’ expectations on code com- pletion and whether existing research has met their demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' To fill the gap, we perform an empirical study by first interviewing 15 practitioners and then surveying 599 practitioners from 18 IT companies about their expectations on code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We then compare the practitioners’ demands with current research via conducting a literature review of papers on code completion published in premier publication venues from 2012 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Based on the comparison, we highlight the directions desirable for researchers to invest efforts towards developing code completion techniques for meeting practitioners’ expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Index Terms—code completion, empirical study, practitioners’ expectations I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' INTRODUCTION Code completion aims to predict the code tokens or lines that programmers tend to input in their daily programming ac- tivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' It can substantially save keystrokes and improve cod- ing efficiency for programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As reported by [1], with code completion, programmers can averagely save 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='65 keystrokes for completing a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Code completion has become the most frequently-used feature of modern integrated development en- vironments (IDEs) [2], and arouse increasingly more attention from both academia and industry [3]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Traditional code completion techniques [14]–[16] adopt static analysis and provide completion candidates according to pre-defined rules, which generally require non-trivial manual efforts and perform ineffectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' To automate the whole process, machine learning (ML)-based methods [17]–[19] such as N-gram models are proposed to learn the code statistics, and thereby can generalize to new code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' With the development of deep learning (DL), many DL-based approaches [3], [20] have been proposed to implicitly learn the patterns in code and achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Some of the DL-based techniques [5], [8], [10], [21] employ pre-trained language (PLMs) such as BERT and GPT [22]–[24] for code com- pletion, since PLMs are trained on large unlabelled corpora and can encode amount of code knowledge into large-scale parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides accurately predicting the next tokens, PLM-based methods can also perform well in predicting the next lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Despite numerous research efforts on code completion, unfortunately, no prior studies have investigated practitioners’ expectations on the techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' It is unclear whether practi- tioners appreciate the current code completion techniques and what aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', completion efficiency and effectiveness) they care most about during adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The practitioners’ perspective is important to unveil critical problems and can provide guidance for software engineer researchers to create solutions that satisfy programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this paper, we follow a mixed-methods approach [25], [26] to gain insights into practitioners’ expectations on code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We start with semi-structured interviews with 15 professionals who have an average of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='4 years of software programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Through the interviews, we qualita- tively investigate the state of code completion practices, issues faced by our interviewees when using code completion tools, and their expectations on code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We then perform an exploratory survey with 599 professionals from 18 IT companies to quantitatively validate practitioners’ expectations uncovered in our interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We finally conduct a literature review of research papers published in premier venues from 2012 to 2022 and compare the techniques proposed in the papers against the expectations of practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Specifically, we investigate the following four research questions: RQ1: What is the state of code completion practices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This research question studies code completion practices , including code completion tools and usage scenarios (the scenarios practitioners use code completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For completion tools, 72% and 54% participants express that they often use code completion in daily programming and are eager for a better code completion tool, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Meanwhile, built- in tools in IDEs are far more popular than third-party plug-in completion tools such as Copilot [11] and IntelliCode [8], used arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='03846v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='SE] 10 Jan 2023 by 96% and about 13% of the participants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For usage scenarios, to facilitate the investigation, we broadly divide code completion techniques according to the completion granularities, including token level and statement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' On average, 81% participants adopt token-level comple- tion, which is evidently more than those adopting statement- level completion (only 32%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For token-level completion, 84% and 85% participants often utilize tools to complete identifiers and recommend APIs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For statement- level completion, completing the currently edited line and predicting the API argument are the two most popular usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ2: Is code completion important for practitioners, and what are the issues?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this research question, we investigate how practitioners perceive the importance of code completion, and study the issues faced by practitioners during programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the im- portance perceived by the practitioners, 87% of them strongly agree or agree with the importance of code completion during software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the issues of code completion, only 36% of the participants think current code completion tools are satisfying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 56% and 58% participants consider erroneous completion and painful tool installation as the main issues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ3: What are practitioners’ expectations on code com- pletion tools?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This research question focuses on practitioners’ expectations on different granularity level code completion tools, including token-level and statement-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We investigate the expectations from multiple aspects including usage scenar- ios, evaluation metrics, access to service (online or offline), completion effectiveness, and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' On average, 88% and 57% participants expect token-level completion scenarios and statement-level completion scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For token- level completion, about 90% participants expect tools to com- plete identifiers, recommend APIs, and complete paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', when the required tokens are related to a file/directory path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 79% of participants are satisfied if the completion tools pro- vide appropriate tokens within the top three candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Most participants (85%) expect tools to generate completion results within 200 milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For statement-level completion, the most expected scenarios are skeleton completion (predicting the skeleton of classes and methods), API argument rec- ommendation (recommending the arguments of called APIs) and completion of currently edited line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The most favored completion latency is no more than 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For evaluation metrics, most participants care about the completion accuracy (79%) and grammatical correctness (86%) for adopting token- level and statement-level completion techniques, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ4: How well current research satisfies practitioners’ demands?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This research question investigates the code completion research and explores the gap between the research and practitioners’ expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We have identified 26 papers about code completion techniques, among which 17 papers propose token-level completion and 10 papers are about statement- level completion1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the research about token-level com- pletion, most of them focus on API recommendation and identifier completion scenarios, which is consistent with prac- titioners’ expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, no papers have covered the path completion scenario, a scenario expected by the large majority of practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the research about statement- level completion, most papers focus on next line completion, but only a few papers explore skeleton completion and API argument prediction that practitioners expect most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, most papers measure overlapped n-grams and edit similarity between completed code and human-written code, however, they are not preferred by the majority of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Our research aims at providing the future research direction on code completion and helping researchers to consider the expectations of practitioners when studying code completion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We hope researchers to propose better code com- pletion tools, which can be eventually adapted into program- ming practices and satisfy a wide range of programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In summary, we make the following contributions: We are the first to shed light on the practitioners’ expec- tations on code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We first interview 15 and then survey 599 practitioners from 18 IT companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We investigate their opinions on code completion tools, the scenarios they use for code completion, and their expectations on code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We comprehensively review the papers published in the premier venues in software engineering and artificial intelligence fields in the last ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Then we compare the published papers with the practitioners’ expectations, highlighting the aspects desirable for researchers to be improved in code completion for meeting the demands of practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Paper Structure: Section II describes the methodology of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Section III presents the results of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We discuss the implications of our results in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Section V discusses related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RESEARCH METHODOLOGY In this section, we introduce our overall research method- ology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Our research consists of three main stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Stage 1: Interviewing with professionals about their practices on code completion, the issues they have met, and their expectations on code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Stage 2: Performing an online survey to validate and extend practitioners’ expectations based on the interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Stage 3: Conducting a literature review to analyze whether and to what extent current techniques have satisfied practitioners’ demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The interviews and survey were approved by the relevant institutional review board (IRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Stage 1: Interview 1) Protocol: Two authors conduct a series of in-person, semi-structured, and in-depth interviews based on an inter- view guide to explore the participants’ practices, issues, and 1The method proposed in [10] can conduct both token-level and statement- level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 1: Example for illustrating the API argument prediction scenario in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' expectations on code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We invite 15 programming professionals from seven IT companies worldwide to partici- pate in the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Each interview lasts 45-60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In each interview, we first ask the interviewees some demo- graphic questions about their background including job roles and working experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Then, we ask the interviewees to freely talk about what they regard as a good code completion tool and their expectations of code completion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In the end, we investigate their code completion practices and the issues they have ever met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) Interviewees: We invite interviewees from our networks in the IT industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The interviewees are working full-time in different roles such as developers and AI algorithms designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Eventually, 15 interviewees from seven IT companies agree to participate in the interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The programming experience of our interviewees varies from 5 years to 13 years, with the average programming experience at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='4 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 3) Data analysis: The first author transcribes the interviews and summarizes the interviewees’ perspectives into opinion cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Then another author verifies the summarized opinion cards and provides suggestions for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' After incor- porating the suggestions, the two authors separately analyze and sort the opinion cards into potential descriptions for the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The Cohen’s Kappa value between the two authors is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='67, indicating a substantial agreement between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The two authors discuss their disagreements to reach a common decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' To reduce the bias of the two authors in sorting descriptions, another two authors have also reviewed and confirmed the final set of survey descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' At the end of the interview, we derive four code completion practices and six issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We summarize three and six usage scenarios for practitioners’ expectations on token-level and statement-level code completion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Furthermore, we derive eight and thirteen factors which specifically affect the adoption of token-level and statement-level completion, respectively, and draw another three aspects of general code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Stage 2: Online Survey 1) Survey Design: The survey consists of different types of questions including single/multiple-choice, Likert scale (Op- tions: Strongly Disagree, Disagree, Neutral, Agree, Strongly Agree), and short answer questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For each question, we add an “I don’t understand” choice to filter the cases that participants do not understand our descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The survey consists of six parts: Demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The survey first asks for demographic information of the participants, including primary job roles and programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We ask the participants about the factors about code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In particular, we investigate the code completion tools they have ever used and their practices on using code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Code completion usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This part aims at in- vestigating in what scenarios practitioners use code com- pletion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We summarize three scenarios for token- level completion and six scenarios for statement-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For each scenario, we utilize a screenshot or GIF for illustration, with an example shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The three token-level completion scenarios include identifier completion, API recommendation, and path completion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', completing the path of a referred file or directory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The studied statement-level scenarios include next line prediction, completion of currently edited line, API argument recommendation, string completion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', completing a log), skeleton prediction and block content prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', predicting a method block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Tool importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We investigate the practitioners’ at- titudes towards the code completion tools in this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We ask the participants about how they perceive the importance of the tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Code completion issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This part investigates the issues faced by the participants during using the code comple- tion tools, such as erroneous completion, high completion latency, high resource consumption and concern about data leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Practitioners’ expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this part, we study prac- titioners’ expectations on code completion tools of the two granularity level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', token-level completion and statement-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For each granularity level, we ask multiple aspects of the expectations including usage scenarios, evaluation metrics, access to service (online or offline), completion effectiveness and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, we also investigate general aspects that affect practition- ers’ likelihood to adopt code completion tools such as time consumption of tool installation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' At the end of the survey, we allow our participants to choose to freely provide their comments, advice, and opinions about code completion and our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' To check participants’ perceptions about the survey length and clarity of the descriptions, we conduct a preliminary survey with a small set of practitioners that are different from our interviewees and survey participants before large- scale distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Based on the received feedback, we make minor modifications to the survey and produce a final version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We utilize widely-used questionnaire websites [27], [28] to distribute the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) Participant Recruitment: We contact professionals work- ing full time in IT companies in our social networks, and ask for their help to complete and disperse the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In particular, we send invitations to the professionals working in Microsoft, Intel, Tencent, Alibaba, and other IT companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 1 from sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='ensemble import RandomForestclassifier 2 X_train = [1,2,3,4] 3 Y_train = [1,0,1,0] 4 model = RandomForestclassifier(n estimators=2) 5 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='fit(X_train, Y_train)TABLE I: Participants roles & working experience Role Population <1y 1-3y 3-5y 5-10y >10y Development 345 32 102 121 65 15 Algorithm Design 173 10 46 81 42 4 Testing 27 6 8 4 6 3 Architect 4 0 1 2 1 0 Project Manager 2 0 0 0 1 1 Others 48 9 15 17 7 0 599 57 172 225 122 23 We finally receive 611 survey responses in total with the average completion time at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Among them, we dis- card 12 responses that are completed within two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The survey results presented in this paper are analyzed from the remaining 599 valid responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' An overview of the surveyed participants’ roles and their experience is depicted in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Most participants are engaged in software development and have 3-5 years of programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 3) Result Analysis: We analyze the results based on the question types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For multiple-choice and single-choice ques- tions, we report the percentage of each selected option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For Likert-scale questions, we draw bar charts to illustrate the distributions of the Likert scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the open-ended short answer questions, we conduct a qualitative analysis of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, we drop “I don’t understand” ratings that form a small minority (less than 1%) of all the received ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Stage 3: Literature Review The papers about code completion are usually published in software engineering and artificial intelligence fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' There- fore, we go through research papers published in ICSE, ESEC/FSE, ASE, ICPC, SANER, MSR, ICSME, PLDI, OOS- PLA, TSE, TOSEM, EMSE, ACL, EMNLP, NAACL, IJCAI, ICLR, NeurIPS, and AAAI from 2012 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We select papers from the above conferences and journals because they are premier publication venues in software engineering and artificial intelligence fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We first read the title and abstract of the papers to check whether they are related to code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For each code completion paper, two authors read its content and analyze the capabilities of the proposed approach in terms of completion granularity, scenarios, evaluation metrics, and the access to the completion service, and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, Izadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [10] declare that they utilize multi-task learning to train the model to predict next token and next statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Thus, we infer that this paper works on both token-level and statement- level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Svyatkovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [3] claim to provide web completion services and client completion model, and we infer the access to its service to be both online and offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Two authors discuss the differences in the capability analysis and confirm the final results through further paper reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RESULT ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ1: Code completion practices In this research question, we investigate the practitioners’ code completion practices, including the used code completion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='0 Others AiXcoder Kite Tabnine Copilot IntelliCode Compiler-based tool IDE built-in tool (a) Statistics of code completion tools that practitioners have used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 10% 20% 30% 40% 50% 60% 70% 80% Percentage of Valid Responses I am eager for a better code completion tool [P3] I find programming laborious without code completion tools [P2] I often use code completion tools [P1] Strongly Disagree Disagree Neutral Agree Strongly Agree (b) Usage status of code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2: Code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' tools and usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The participants’ rating results for some descriptions related to code completion practices are illustrated in Figure 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 1) Code completion tools: We first ask the participants what code completion tools they have used and the results are shown in Figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From the results, we find that 96% participants express they adopt IDE built-in code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Compiler-based tools such as CCLS and Clangd [14] are the second most popular tools among surveyed practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, the third-party plug-in tools are not prevalent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', about 13% participants have used Copilot [11] and IntelliCode [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We then investigate the practitioners’ usage status of code completion tools, with results illustrated in Figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Al- though most practitioners (72%) indicate that they often adopt code completion in daily programming, only 50% participants agree that they find programming laborious without code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 54% of them are proactive in seeking a better code completion tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' “I always keep tuned on the release of new code completion tools and am eager to be the first to use them.”, as a participant stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 1: The most commonly-used code completion tool is IDE built-in tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Other third-party plub-in tools such as IntelliCode [7] and Copilot [11] are far less popular, for which only about 13% participants have used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In addition, most participants express that they often use code comple- tion tools, and 54% of them are eager for a better code completion tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) Usage scenarios: In this section, we investigate in what scenarios the practitioners use code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' According to [29], we study the scenarios from two granularities in- cluding token-level and statement-level completion, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Token-level completion aims at completing the currently typing token and the next token to be used for developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Statement-level completion is able to predict multiple tokens and even lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We provide the definitions of the two granularity level completion in the survey for facilitating participants’ understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The token-level completion and 0% 10% 20% 30% 40% 50% 60% 70% 10% 20% 30% 40% 50% 60% 70% 80% 90% Percentage of Valid Responses Block content prediction [C9] Skeleton prediction [C8] String completion [C7] API argument recommendation [C6] Completion of currently edited line [C5] Next line prediction [C4] Path completion [C3] API recommendation [C2] Identifier completion [C1] Token Statement Strongly Disagree Disagree Neutral Agree Strongly Agree Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 3: Scenarios that practitioners use code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% Percentage of Valid Responses Code completion improves programming experience [M5] Code completion can reduce my typing mistakes [M4] Code completion can provide some hint [M3] Code completion can save my key strokes [M2] Code completion can improve my efficiency [M1] I think code completion is important Strongly Disagree Disagree Neutral Agree Strongly Agree Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 4: Importance of code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' statement-level completion involve three and six scenarios, respectively, as introduced in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The participants’ ratings of the usage scenarios are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We ob- serve that the overall popularity of statement-level completion scenarios is obviously less than that of token-level (32% v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 81% on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Token-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As can be seen in the upper part of Figure 3, all the three usage scenarios are popular among the practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Specifically, about 85% participants use code completion to complete identifiers and recommend APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' More than 70% survey practitioners adopt code completion for path completion in their daily programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 2: All the three token-level code completion usage scenarios are popular among the participants, among which identifier completion and API recommendation are the most widely used scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Statement-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As shown in the lower part of Figure 3, the three most popular statement-level completion scenarios are API argument recommendation, completion of currently edited line, and skeleton prediction, used by 46%, 45%, and 36% participants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In addition, about 25% of them adopt code completion to predict the next line of code and string content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Only 16% of survey participants adopt code completion in the block content prediction scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One participant said that “The performance of existing completion tools is not good enough to predict the whole block content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The predicted results may be severely misleading in some cases, which reduces my programming efficiency.” Finding 3: The most adopted statement-level code comple- tion scenarios are API argument recommendation, comple- tion of currently edited line, and skeleton prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 10% 20% 30% 40% 50% Percentage of Valid Responses Painful tool installation [I6] Erroneous completion [I5] Concern about data leakage [I4] High resource consumption [I3] High completion latency [I2] Unsatisfying ranking [I1] The tools are satisfying for me Strongly Disagree Disagree Neutral Agree Strongly Agree Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 5: Issues of code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ2: Code completion Importance and Issues 1) Importance of Code Completion: We ask whether they think code completion is important for development, and their agreements on the possible reasons of the importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The results are illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From the results, we can find that most participants (88%) think that code completion plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 90% of them agree or strongly agree that code completion can provide some hint on the implementing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One practitioner described that “Sometimes I forget how to type a long identifier or method name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this case, code completion can show me the candidates and provide some hint for completing the code.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, more than 85% of them think completion tools are helpful to reduce typing mistakes, improve programming efficiency, save keystrokes, and improve programming experi- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 4: 88% of the survey participants agree that code completion tools play an important role in programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Most participants (90%, 87% and 86%) think the tools can provide some hint, improve programming efficiency, and improve their programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) Code completion issues: Figure 5 illustrates the partici- pants’ ratings of code completion-related issues they faced dur- ing programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Despite the importance of code completion, only 36% participants think the tools they are currently using are satisfying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 45% and 44% participants consider erroneous completion and painful tool installation as the main issues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Erroneous completion indicates that some errors occur in the completed code, which practitioners need to modify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As a participant said: “The completion is usually not exactly what I need, and it is annoying for me to rectify the errors.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 34% and 31% of them are not satisfied with the rankings of appropriate tokens in the candidates and concerned about data leakage in the tools, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Among the participants, about a quarter of them hold the view that the high resource (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', memory and CPU) consumption and high completion latency are not acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' A participant working on large C++ projects told us: “The C++ code completion tools like CCLS and Clangd parse the opened cpp files and build indexes for them when programming, which may consume all of my CPU and memory.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 5: The most concerned issues by practitioners are the erroneous code completion, painful tool installation, and unsatisfactory ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 10% 20% 30% 40% 50% 60% 70% 80% 90% Path completion [T3] API recommendation [T2] Identifier completion [T1] (a) Expectations on usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 10% 20% 30% 40% 50% 60% 70% 80% Recall [T6] Average Rank [T5] Accuracy [T4] (b) Expectations on evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 10% 20% 30% 40% 50% 60% 70% 80% Offline [T9] Internal Network [T8] Public Network [T7] Strongly Disagree Disagree Neutral Agree Strongly Agree (c) Expectations on access to completion service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Rank of the needed token in completion candidates [T10] 1 2 3 4 5+ (d) Expectations on code completion effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Maximum completion latency [T11] 100ms 200ms 300ms 400ms 600ms 800ms 1000ms (e) Expectations on code completion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 6: Expectations on token-level code completion, where horizontal axis denotes percentage of valid responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ3: Practitioners’ Expectation In this research question, we comprehensively investigate the practitioners’ expectations on code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We explore the aspects including usage scenarios, evaluation metrics, access to service, completion effectiveness, and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We report practitioners’ expectations on the two levels of granu- larity of code completion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', token-level and statement-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We also investigate practitioners’ expectations on three other aspects of general completion tools that impact their usage experience, including time consumption of instal- lation, personalized completion and additional information display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 1) RQ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='1 Expectations on token-level completion: The participants’ ratings for the expectations on token-level code completion is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From Figure 6 (a), most participants show positive attitudes towards the three scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' More than 88% of them expect to use code completion in Identifier completion and API recommendation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 86% of partici- pants expect code completion tools to complete the path they are typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In one participant’s opinion, path completion is very important when he works on a large project, because he/she finds it very easy to make mistakes on file names without path completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 6: For token-level code completion, more than 80% participants agree that tools are supposed to support identi- fier completion, API recommendation and path completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We study participants’ opinions on eval- uating token-level code completion tools in Figure 6 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We observe that Accuracy gains the most support rate (79%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, more than 70% of participants also care about Av- erage rank of the needed token in the candidates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', Mean Reciprocal Rank MRR) and Recall of the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Access to service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Figure 6 (c) shows that 82% of partici- pants expect to use token-level completion tools offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One participant stated that: “Offline tools are the most stable ones, which will not be affected by network status and I can use them anywhere.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In contrast, the support rate of online tools is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 62% and 50% of participants are willing to use the online tools in the internal network and in the public network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Figure 6 (d) shows the practitioners’ satisfac- tion ratio against different ranks of needed token in completion candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From our survey, if the code completion tool can rank the needed token in the top 3 candidates, it will satisfy about 80% practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We show the participants seven GIFs with different completion latency including 100, 200, 300, 400, 600, 800, and 1000 milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From Figure 6 (e), we find that if a token- level code completion tool can give completion candidates within 200 milliseconds, it can satisfy 85% practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) RQ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='2 Expectations on statement-level completion: The results of participants’ expectations on statement-level completion can be accessed in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The practitioners’ expectations on the statement-level code completion scenarios are illustrated in Figure 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We observe that practitioners’ expectations show a similar trend with the popularity of usage scenarios as shown in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Likewise, The three most expected scenarios are skeleton prediction, completion of currently edited line and API argument recommendation, where all of them gain at least 60% support rate, and skeleton prediction is the most expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 51% participants agree that statement-level completion tools are supposed to predict block content such as method block and loop block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 53% of them expect to use code completion tools to predict strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Interestingly, though the scenario of predicting the next line of code is kind of similar to completing the currently edited line, only 43% of participants regard next line prediction as an important scenario while the currently edited line completion obtaining a 62% support rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As one participant commented: “When finishing a line of code, I will start to type the next line immediately instead of waiting for completion results at the end of the line.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 7: skeleton prediction, completion of currently edited line and API argument recommendation are consid- ered to be the most expected statement-level code comple- tion scenarios, which are similar as the popularity of the usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 10% 20% 30% 40% 50% 60% 70% 80% Block content prediction [S6] Skeleton prediction [S5] String completion [S4] API argument recommendation [S3] Completion of currently edited line [S2] Next line prediction [S1] (a) Expectations on usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 10% 20% 30% 40% 50% 60% 70% 80% 90% Grammatical correctness [S11] Readability [S10] Function similarity [S9] Edit similarity [S8] Overlapped N-grams [S7] (b) Expectations on evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 10% 20% 30% 40% 50% 60% 70% Offline [S14] Internal Network [S13] Public Network [S12] Strongly Disagree Disagree Neutral Agree Strongly Agree (c) Expectations on access to completion service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Minimum grammatical correctness rate [S16] Maximum manual modification rate [S15] 20 40 60 80 100 (d) Expectations on code completion effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 10% 20% 30% 40% Which one is more important?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Effectiveness or Generated Code Quantity [S17] Effectivess is much more important Effectivess is more important Neutral Generated code quantity is more important Generated code quantity is much more important (e) Effectiveness v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Generated code quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Average time consumption of completing a line of code [S18] <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5~1s 1~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5~2s >2s (f) Expectations on average time consumption of generating a line of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Maximum time consumption of giving completion results [S19] <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5~2s 2~5s 5~10s >10s (g) Expectations on maximum time consumption of giving completion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 7: Expectations on statement-level code completion, where the horizontal axis denotes percentage of valid re- sponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We illustrate the practitioners’ expecta- tions on different evaluation metrics for statement-level code completion approaches in Figure 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The top three preferred evaluation metrics are grammatical correctness, readability and functional/structure similarity, supported by more than 80% participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, the overlapping n-grams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', BLEU score [30]) and edit similarity receive the least support rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', 65% and 70%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Access to service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Figure 7 (c) presents the likelihood of dif- ferent access to use a statement-level code completion tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Al- though the support rate of offline service drops 15% compared to token-level completion (from 82% to 67%), offline tools are still the most expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, the willingness to adopt an online statement-level code completion tool increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, 66% participants agree or strongly agree to invoke online completion services in internal networks (compared with 62% for token-level completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Furthermore, the sup- port rate of public-net tools also increases from 50% to 58%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One participant told us: “Currently almost all statement- level completion tools are AI-based, which means that their computation complexity is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Thus, utilizing online code completion service is acceptable.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, we observe the difference between the participants’ expectations on tools deployed in public and internal network (58% and 66%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' According to our survey, the difference can be attributed to the concern about data leakage and network stability in public network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As a participant stated that in his/her department, Copilot was not permitted to be installed because it would upload their code to the cloud and might result in leakage of confidential code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From Figure 7 (d), we find that only 26% participants can accept the ratio of manual modification greater than 60% of completed code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 35% and 39% of our participants expect the maximum modification ratio to be no more than 20% and 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For grammatical correctness, if a statement-level code completion tool ensures that at least 80% of the completed code is grammatically correct, it can satisfy 82% practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 18% participants have strict requirements on the correctness of generated code, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', they expect the tools to make no mistake on code grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' According to existing work [9], [21], the more code is com- pleted, the more errors occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We further investigate the practi- tioners’ preference between completion effectiveness (generate less code with higher accuracy) and generated code quantity (generate more code with more manual modifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The results are illustrated in Figure 7 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From our survey, there is no very clear winner between them (46% support rate for effectiveness and 40% for generated code quantity), and a few more participants consider effectiveness more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this question, participants mainly have two points of view: 1) Effectiveness is more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the supporters of effectiveness, completion results with errors may affect their efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' A participant shared his/her experience with statement-level code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' “It was very annoying for me to modify the completion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' I’d rather have less code completed than modify completion results.” 2) Code quantity is more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Supporters of code quantity expect statement-level completion tools to pre- dict more code at once, even if the code needs to be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One participant stated that as long as the tool could predict the code with the right logic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', correct code skeleton), the tool was effective in his eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We study practitioners’ expectations on the effi- ciency of statement-level code completion tools from two as- pects including average time consumption of completing a line (shown in Figure 7 (f)) and maximum latency of generating completion results (shown in Figure 7 (g)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From our survey, we observe that the speed of averagely taking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5 seconds to generate a line receives the highest 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% Maximum Installation time consumption [O1] 1min 5mins 10mins 20mins Other (a) Expectations on time consumption of tool installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 0% 10% 10% 20% 30% 40% 50% 60% 70% 80% 90% I hope code completion tools can learn my programming habits [O3] I hope code completion tools can provide additional information [O2] Strongly Disagree Disagree Neutral Agree Strongly Agree (b) Expectations on additional information display and personalized com- pletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 8: Expectations on other aspects of general code com- pletion tools, where the horizontal axis denotes percentage of valid responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' support rate at 44%, and 39% participants expect the time to be no more than 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the maximum completion latency, the practitioners who expect the code completion tools to generate completion results in less than 2 seconds have the largest proportion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', 47%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' If a statement-level code completion tool generates completion results within 2 seconds, it can satisfy 67% practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In addition, few participants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', 4%) express they are willing to wait for more than 5 seconds for completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 8: For completion efficiency, most practitioners expect statement-level code completion tools to take less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5 seconds to complete a line of code on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Be- sides, to meet the majority (67%) of practitioners’ demands, tools are supposed to generate completion results within 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 3) RQ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='3 Expectations on other aspects of general code completion tools: In this section we present the practitioners’ expectations on general aspects of code completion, and the results are illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Figure 8 (a) shows that practitioners’ expectations on the time cost of installing code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 83% participants will be satisfied if they can install the tool within 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, we observe that 14% of participants are willing to spend more than 20 minutes on installing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As one participant said: “I would like to spend a whole day on installing a code completion tool as long as it can really improve my efficiency.” From Figure 8 (b) we observe that most (76%) practition- ers agree or strongly agree that code completion tools are supposed to learn their programming habits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', preference for function names and variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In addition, 83% of them expect code completion tools to simultaneously present some additional information about the predicted code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', showing users the API definition or documentation while recommend- ing an API).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One participant told us: “Sometimes I forgot the order of arguments in the API, thus showing me how the API is defined can greatly reduce my programming mistake.” TABLE II: Capabilities of current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Likert score de- notes the average weighted score of participants’ ratings (1 to 5 correspond to strongly disagree to strongly agree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Description Likert Papers Score Token-level completion Usage Scenarios Identifier completion [T1] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='50 [5], [10], [31]–[35] API recommendation [T2] [3], [17], [31], [36]–[38] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='50 [4]–[6], [32], [33], [35] [7], [10], [19], [34], [39] Path completion [T3] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='39 Evaluation Metrics Accuracy [T4] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='19 [10], [17], [32], [34] Average rank [T5] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='00 [3], [5], [7], [36], [38] Recall [T6] [19], [33], [34], [36] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='05 [7], [31], [35], [39] [3], [6] Others [17], [32], [36], [37] Access to service Online [T7] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='43 [3], [10] Offline [T9] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='20 [3], [7], [17] Time Consumption Mentioned [6], [32]–[34], [38], [39] [3], [7], [10], [17], [36] Statement-level completion Usage Scenarios Next line prediction [S1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='27 [8]–[10], [21] [20], [37], [40], [41] Completion of currently edited line [S2] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='74 [8], [20], [21] API argument recommendation [S3] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='81 [8], [21], [34] String completion [S4] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='32 Skeleton prediction [S5] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='88 Block content prediction [S6] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='43 [21], [40], [42] Evaluation Metrics Overlapped n-grams [S7] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='72 [9], [21], [37], [40] Edit similarity [S8] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='81 [8], [37], [40], [42] Function similarity [S9] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='09 [9] Readability [S10] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='31 Grammatical correctness [S11] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='42 Others [8], [20], [40]–[42] Access to service Online [S12] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='54 [8], [10], [42] Offline [S14] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='89 [41] Time Consumption Mentioned [8], [20], [40], [41] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RQ4: Current State-of-the-art Research After our literary review, we identify 26 papers in total from the top conference and journals in software engineering and artificial intelligence communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Table II shows the capabilities of surveyed code completion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We can observe that the research on token-level completion is much more than that on statement-level completion (17 papers v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 10 papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 1) Token-level completion: Usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As seen from Table II, all of the 17 collected token-level completion papers support API recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, only a few of these papers work on identifier completion, and none of them mention path completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This may be attributed to that most identifiers and path tokens are Out-of-Vocabulary (OoV) words for the proposed techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this part, accuracy is equivalent to top- 1 accuracy and error rate, recall refers to the more general top- k accuracy, and average rank refers to the average rankings of the appropriate token in the prediction candidates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', Mean Reciprocal Rank, MRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In Table II, we find that most of the papers utilize recall and averaged rank to evaluate the effectiveness of their approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, other evaluation metrics are explored [3], [7], [17], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Some researchers attempt to explore better metrics as the proxy for users’ productivity such as saved keystrokes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Access to service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Few papers have focused on employing their techniques in industrial products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We classify the papers into two categories including online and offline according to their deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Svyatkovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [3], [7] develop their system as part of Intellicode extension in Visual Studio Code IDE [43], allowing programmers to use it offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Izadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [10] utilize a Transformer architecture, claiming that they put the model on the cloud and provide completion web services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, Hindle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [17] mention that their tool is incorporated into the offline Eclipse plug-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Among collected token-level code completion pa- pers, 11 of them explicitly consider the time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Nine approaches proposed in [3], [6], [7], [10], [17], [32], [33], [36], [38] take less than 200 milliseconds to predict a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' According to our results, these techniques with such latency can satisfy 85% practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, another two methods take several seconds to complete a single token [34], [39], which is considered unacceptable for token-level completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) Statement-level completion: Usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' As seen from Table II, most papers focus on next line prediction which receives only 43% support rate, but the more expected scenarios completion of currently edited line and API argument recommendation obtain less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Three papers propose approaches for block content prediction scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, no papers propose techniques to predict the methods’ skele- tons (skeleton prediction) and complete string content (string completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 9: Most papers focus on next line prediction scenario, while only a few of them explore completion of currently edited line and API argument recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, no paper has proposed techniques for skeleton pre- diction, which is considered as the most expected scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From Table II, most papers evaluate the generated code via overlapped n-grams such as BLUE and ROUGE, and edit similarity, which receive the least support rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', 65% and 70%, respectively) according to our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Only one paper evaluates the generated code against human- written code via function/structure similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, none of them mention the readability and grammatical correctness of the generated code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Some papers also propose customized metrics to comprehensively evaluate their statement-level com- pletion approaches such as click-through rate [8], [20], [40]– [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 10: Most papers focus on measuring overlapped n-grams and edit similarity between the generated code and the human-written code that are not preferred by the large majority of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' No paper evaluates the generated code via grammatical correctness and readability, which the practitioners value most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Access to service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The work [8] proposes a tool called Intellicode Compose, which deploys the models on the cloud and also allow client-side caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [42] present an android studio plugin as a web service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In addition, the tool proposed in [41] serves as an Eclipse plug-in and provide offline completion service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Among the collected papers that propose statement-level code completion approaches, only four of them explicitly discuss the time consumption of their techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For example, the methods proposed in [8], [21], [41] take less than 1 second to complete code statements, while the work in [20] takes more than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='5 seconds on average for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Finding 11: Time consumption, a critical adoption factor of code completion tools, is missing in 40% of our collected token-level and statement-level papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Implications Our survey results highlight several implications for the research community: 1) Implication on code completion scenarios: For token- level code completion tools, besides identifier completion and API recommendation, they are also expected to support path completion which is expected by 86% participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For the code completion tools that are capable of predicting lines of code, the most important scenarios for programmers are skeleton prediction, completion of currently edited line and API argument recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, most current papers about statement-level completion focus on less anticipated scenarios such as next line prediction [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 2) Implication on code completion tools: The large ma- jority of programmers expect code completion tools to be more intelligent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, 76% participants expect the tools to learn their programming habits so that they can keep the programming style consistent and reduce modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, most of them are also eager to be informed with additional information about the predicted code such as API definition or documentation (81%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' One participant stated that “It is hard for me to remember countless APIs in my project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' When I use an API, if the code completion tool locates and shows the API documentation simultaneously, it can significantly save time in searching the API.” Besides, most practitioners wish the time consumption for tool installation and configuration to be within 20 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' If they have to spend much time on tool installation, they may be dissuaded from using it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 3) Implication on evaluation metric: Evaluation metric is another important factor that should satisfy practitioners’ expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Most of existing studies about statement-level code completion evaluate the generated code by comparing with human-written code in terms of overlapping n-grams (such as BLEU score and ROUGE) [21] and editing sim- ilarity [29], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, these two metrics receive the least support rate among all the evaluation metrics in our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Practitioners more expect the tools to use the metric function/structure similarity for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' A participant told us: “ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' BLEU score may be a good criteria to judge the quality of generated natural language texts, however, it can hardly evaluate whether a code snippet is satisfactory.” In addition, the metrics that participants value most such as grammatical correctness and readability of generated code are missing in current publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 4) Balancing effectiveness and code quantity: From our survey, 46% practitioners regard the effectiveness of statement- level code completion weighs more than the quantity of pre- dicted code (39%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Considering that two completion strategies are both preferred by a certain number of participants, it will be better if the tools provide a configurable option for users to decide the quantity of predicted code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, we also observe that most practitioners expect that the completed code does not need extensive manual modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, 74% participants cannot accept that more than 40% of completed code needs manual modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Thus, code completion tools may preliminarily estimate the probability of whether over 40% of predicted code needs modifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' If the probability is high, tools can terminate current completion process and generate completion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 5) Implication on code completion latency: Completion la- tency is one key aspect that substantially affects practitioners’ likelihood to adopt code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From our survey, only 15% participants can accept that tools take more than 400 milliseconds to predict a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Besides, 83% of them wish the average time consumption of generating a line of code to be less than 1 second, and 80% participants expect the statement-level completion latency to be no more than 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, the time consumption factor is missing in nearly half of our collected papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Researchers should pay more attention on code completion latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Considering that offline tools are the most expected, how to effectively compress the completion models is an important direction for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' 6) Improving robustness of code completion tools: In our survey, many participants mention that the robustness of code completion tools is also vital to user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Robustness requires that the completion results are not affected by slight perturbations in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' From our survey, a participant shared his/her experience with Tabnine: “Tabnine can suc- cessfully predict the API arguments if the variable is named ‘X train’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, if I modify the name to ‘x train’, the completion results will be totally different.” Besides identifier changes, code completion tools are also supposed to be robust to statement changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' A participant stated that: “AiXCoder can predict the block content of a method well, but the results turn to be terrible when I inserted an unrelated assignment statement.” B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Threats to Validity One of the threat in our survey is that there may be some participants who do not fully understand the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, some participants have never used statement-level code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Therefore, they may be unfamiliar with the questions of statement-level code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' To reduce this threat, we utilize one or two clear images or GIFs to describe each scenario and facilitate them better understanding the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Furthermore, some participants do not answer the questions seriously and the results cannot reflect their beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Therefore, we drop the responses completed less than two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' This is a common and tolerable threat to validity in previous studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=', [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Another threat is that our participants may not be represen- tative in typical programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Our solution is to widely survey practitioners working in many IT companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We believe we have made this threat have minimal impact on the results of our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Code Completion Traditional code completion focused on static analysis tech- niques associated with manually defined rules to suggest code [15], [16], [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, researchers utilized type information [16], [47], similar code snippets [48] and history data [49] to predict needed code tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Equipped with machine learning, a series of code com- pletion work equipped with statistical language models was proposed [17]–[19], [34], [36], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, Hellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [36] explicitly took the techniques such as nested scopes into account and improved the performance of the n-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, Raychev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' combined decision trees and domain knowledge, proposing a probabilistic model [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' With the development of deep learning, neural networks such as RNN [50] and Transformer [51] showed great ca- pability to learn features from source code [5], [31], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [31] proposed a point mixture network to relieve the Out-of-Vocabulary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [5] parsed the code into abstract syntax trees and fed the trees into a Transformer-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In recent years, pre-trained language models have been leveraged for predicting multiple code tokens [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Svyatkovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [8] proposed GPT-C, which could predict code statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Izadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [10] raised CodeFill, which combined type and semantic information of code, further improving the completion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Studies on Code Completion Practices Apart from research on code completion, some other work focus on studying code completion practices [52]–[55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' For instance, Vaithilingam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [52] asked participants to fin- ish programming tasks with or without Copilot [11] and determined whether Copilot is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In addition, Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' [53] focused on investigating evaluation metrics of code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Authors compared the measurable user data (objective data) and the user-reported productivity (subjective data), and identified the most representative metric as a proxy of productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' The work [54], [55] both pointed out the differ- ence between synthetic data and real-world data, and identified the most important tokens that needed to be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' However, no prior studies have investigated the practi- tioners’ expectations on code completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' In this paper, we conduct a large scale user study, investigating practitioners’ expectations on multiple aspects of code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Moreover, we perform a comprehensive literature review to reveal the gap between current techniques and practitioners’ expectations, providing future directions for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this paper, we interview 15 professionals and survey 599 practitioners on completion practices, issues they face and their expectations on code completion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Practitioners expect code completion tools to suggest code for different granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' Practitioners also expect a code completion tool to satisfy the aspects including usage scenarios, evaluation metrics, access to service, effectiveness and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' We also compare the capability of current research with practitioners’ expectations via a literature review, pointing out the aspects to be improved for meeting the demands of practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE2T4oBgHgl3EQfXgf7/content/2301.03846v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': 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Being able to operate autonomously. As IoT +devices have become more prevalent, they have become the most tempting targets +for malware. Weak, guessable, or hard-coded passwords, and a lack of security +measures contribute to these vulnerabilities along with insecure network +connections and outdated update procedures. To understand IoT malware, current +methods and analysis ,using static methods, are ineffective. The field of deep +learning has made great strides in recent years due to their tremendous data mining, +learning, and expression capabilities, cybersecurity has enjoyed tremendous growth +in recent years. As a result, malware analysts will not have to spend as much time +analyzing malware. In this paper, we propose a novel detection and analysis method +that harnesses the power and simplicity of decision trees. The experiments are +conducted using a real word dataset, MaleVis which is a publicly available dataset. +Based on the results, we show that our proposed approach outperforms existing +state-of-the-art solutions in that it achieves 97.23% precision and 95.89% recall in +terms of detection and classification. A specificity of 96.58%, F1-score of 96.40%, an +accuracy of 96.43. + + +Keywords: IoT-Malware; malware detection; decision trees; deep learning, anomaly detection with +decision trees. + + + +1. Introduction +An intrusion is an attempt to compromise security objectives by infecting a system. +Therefore, many tools and methods have been developed to protect networks and +systems from intrusion, such as detection systems [7–9]. Thus, intrusion detection +consists of techniques that classify data activity into normal and intrusive [6, 8] to +detect undesirable activity. An intrusion detection system detects and stops intruders +from entering a monitored network outside or inside. Two methods of detection are +generally used for this purpose. For example, misuse detection detects intrusion by +using a known attack signature. Another detection method is anomaly detection, +which is based on deviations from a normal model [1, 8, and 10]. By combining +misuse detection with anomaly detection, hybrid detection approaches are aimed at +increasing the detection rate and accuracy of IDS [9, 11, and 12]. + + + + + + +These IDS are efficient, but they have a lot of limitations, like real-time detection, alarm generation, and +data accuracy, which can lead to less successful detections [6, 8]. Because of this, intrusion detection is still an +important and dynamic research field. We've been integrating ML methods to improve intrusion detection and +strengthen computer security. Several papers have explored using machine learning to enhance data quality +and training to improve intrusion detection performance [13–20]. There are many issues in which decision trees +have been used for classification. You test each feature individually. After each branch is split, a single +classification is assigned to it [21, 22]. It represents the training set better than a decision tree. It can predict +their values because it includes the values of instances outside the training set. There is a guarantee that the +decision tree constructed by ID3 and C4.5 will correspond to the data provided by both of these well-known +algorithms. +Meanwhile, the data are not always gathered in a structured manner. Unstructured data must +be preprocessed before they can be analyzed. Additionally, selecting relevant features is an +important step aimed at reducing the computational costs of modelling and improving the +predictive model's performance [13, 24]. +An approach based on decision trees for network intrusion detection and making accurate +decisions is proposed in this paper. The quality of the data was improved through the use of +feature engineering. We have validated two major contributions in this study. The first step is to +improve data quality by applying the entropy decision method. We also developed a classifier +model based on decision tree algorithms to detect network intrusions effectively. Please find the +following section for more information. +Related Works +Over the last decade, some intrusion detection techniques have been adopted to achieve +computer security objectives. The goal of intrusion detection research is to increase the + +effectiveness and capability of IDS through automatic responses. ML techniques are becoming +more widely used in intrusion detection [13–20]. As such, intrusion detection using ML is a +classification project involving using labelled data to build classifiers that can determine the +difference between normal and abnormal activity [11, 16, 21, 27, and 28]. They allow an +effective classifier to be trained and built using relevant data [13, 17, 23, 25, 35, and 36]. + An anomaly-based intrusion detection system based on fuzzy SOM method was proposed +by Karami [37] in 2018. A model for intrusion detection combining NB and DL technique was +proposed by Tabash et al. [26] in 2020. A genetic algorithm was used in the model for feature +selection. The detection model proposed by Ghazali et al. [27] for intrusive communication was +published in 2015. These methods include Simple Cart, NB, BFTree, PART, and Ridor. For +network intrusion detection, It achieved an accuracy of 89.24% using NSL-KDD dataset. +Using the random forest algorithm, Hadi [29] proposed a model for selecting significant +features in 2018. NSL-KDD was used to evaluate this model. According to this model are all +within the CPI range. As a method to improve data quality, Gu et al. [17] proposed ensemble +SVM-based intrusion detection with LMDRT transformations in 2019. Results show that the +class performance on CICIDS2017 dataset is ACC 93.64%, DR 97.56%, and FAR 20.28%. A +double PSO metaheuristic DL model was developed by Elmasry et al. [32] in 2020 to detect +network intrusions. ACC, DR, and FAR values are generated on the CICIDS2017 dataset. +According to Prasard et al. [36], a new IDS was proposed in 2019 that uses a probabilistic +method in order to extract significant features from a subset of features. +A state-of-the-art literature review demonstrates that the learning methods and the data +quality are important factors determining the robustness of IDS [6, 17, 26–29, 32, 36, 37]. +Malware attacks can be detected and prevented using several methods. Systems for detecting + +and preventing incidents by intruders (IDPS) use several techniques. The data mining approach +includes signature-based technologies, stateful protocol analyses, behavioral analyses and +anomaly-based techniques. Different detection approaches are used in these technologies. +Signature-based detection is an effective method for detecting already known threats but is not +very effective when detecting previously unknown risks. +Similarly, they cannot be used to detect disguised threats when using evasion. On the other +hand, anomaly-based screening can detect an invasion without requiring signatures. An unknown +intrusion can instead be spotted by observing similar behaviour in other intrusions. Malware +occurrences are identified through the concept of modelling normality associated with anomaly- +based detection. Observing different activities and characteristics results in the creation of these +sketches. It can uncover previously undiscovered threats because anomaly-based detection looks +for abnormal patterns. +1. Novel Network Intrusion Detection Approach +3.1 A Proposed Model. This model comprises three main components, as shown in Figure 1: data +quality, the building of a classifier, and deployment of intrusion detection. Here are details on +what each of these components includes. +Part 1: Process of Data Quality. +Among its primary responsibilities is the collection and preprocessing of data. As a +result, the system follows a process that gathers and accumulates data from networks. +Following data collection, the gathered network traffic is subjected to data preprocessing. +Preprocessing treats the data type incompatibilities and ignores them. A further step +involved sanitizing the data and saving the results. The data is further transformed, +resulting in the finalized features of the network dataset. These features are chosen using + +the entropy decision procedure. +Part 2: Classifier Construction. +The second part will be started after the first has been completed. As the title indicates, +the goal of the second component is to create a classifier model. Input is the data that was +transformed in the data quality process. There are two main phases of classifier +construction: training and validation of the model. The first phase of our proposed +method involves training a decision tree classifier with three portions of data. Our model +is then validated using the rest of the data in the second phase. +Part 3: Defining network intrusion detection systems. +For effective IDS to be improved, actual testing is necessary. Therefore, we can test its +ability to differentiate between normal and abnormal behaviours. Thus, the IDS can be +made more accurate based on classification results. To detect anomalous behaviour in +systems, various approaches have been utilised in the literature. Researchers are very +interested in intrusion detection and prevention since there is a concern about protecting +systems from intrusion while maintaining network security regulations. This section +describes the various strategies employed in prior research for intrusion detection and +prevention. +3.2. Description of Proposed Solution. We have explained above that our approach begins with +collecting and transforming data, followed by selecting features based on the needs of analysis +and detection. Creating an intrusion detection model that is accurate requires good quality data. +Thus, this step focuses on preparing data for analysis and making accurately informed decisions. +To obtain a good training set, we first perform feature extraction using entropy decision on +original traffic inside network traffic. It is a crucial step aimed at increasing the accuracy of our +methodology. Besides reducing training complexity by analyzing less data, the goal is to achieve + +a great model that performs well in accuracy, detection rate, and real-time detection. Before +analyzing network traffic, we apply specific preprocessing to collected data. It is then +normalized. This process reduces the number of features in collected network data. +To improve our approach accuracy, we implement proposed data quality techniques to +transform the data. Therefore, an effective intrusion detection model can be trained and validated +by using the decision tree to make timely decisions. Furthermore, intrusion detection is +Cybersecurity is increasingly becoming an established topic with the emphasis on +securing sensitive data stored in computer systems and networks. By blocking unauthorized +access to the system, attackers can't obtain, corrupt, damage, or destroy this information. A +computer intrusion detection system identifies any breach of security in a computer network by +revealing activity in the network. Detecting potential malware and its ability to cause harm is the +purpose of an intrusion detection system (IDS). We must recover from malware and detect it at +this point. An intrusion prevention system (IPS) detects malware in operations and also tries to +prevent the occurrence of potential incidents. Although the internet has changed a lot of things, it +has also presented a number of insecurity areas that need to be addressed in a secure environment +to be maintained. + + + + + + + + + + + + + + + + + + + + + + +Figure 1: Proposed network intrusion detection model. +Input Network Traffic +Collection and Pre- +processing +Entropy Decision +Feature Selection +Data Quality +Part + +Transformed Data +Training of Decision Tree +Model +Validation and Building +of +Decision Tree Classifier +Classifier Part +Input Network Traffic +Test Traffic + +Network Intrusion +Detection +Aprocach +Intrusion +Detection +Deployment +Part + +Sensors 2022, 22, 4302 +2 of 15 + +Experimental Results and Discussion +Validating intrusion detection programs rely heavily on the evaluation of datasets. As a result, to evaluate +an IDS using ML techniques, many available and appropriate datasets are available. We have chosen two types +of datasets to evaluate and validate our approach, including NSL-KDD and CICIDS2017. The NSLKDD +dataset is very practical because of its novelty and the sheer number of examples it contains. CICIDS2017 was +created from a dataset created by Canadian Institute for Cyber Security. Presented here is an effective intrusion +detection dataset that overcomes the limitations of the actual dataset. +Table 1. Hyperparameters deployed for model best performance. + + Hyperparameters +Values +Batch size +64 + Number of Epochs +25 +Learning rate +1e-3 + Optimization Algorithm +Adam +Loss function +Cross-entropy + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Sensors 2022, 22, 4302 +3 of 15 + +Table 2. Shows the distribution of our Malevis dataset. + + + + + + + +Category +Type +Class +Samples +Total +Benign +Normal +1832 +1832 +Adposhel +494 +Amonetize +497 +BrowseFox +493 +Adware +2983 +InstallCore +500 +MultiPlug +499 +Neoreklami +500 +Agent +470 +Dinwod +499 +Elex +500 +HackKMS +499 +Trojan +Injector +495 +4440 +Regrun +485 +Malware + Snarasite +500 +VBKrypt +496 +Vilsel +496 +Neshta +497 +Sality +499 +Virus +1997 +Expiro +501 +VBA +500 +Allaple +478 +Autorun +Worm +496 +Fasong +1974 +500 +Hlux +500 +Androm +500 +Backdoor +1000 +Stantinko +500500 +500 +500 +500 +1832 +494 +496 +497 +478 +500 +493 +501 +500 +499 +499 +500 +497 +496 +470 +496 +499 +500 +500 +485 +499 +Normal +Adposhel +Amonetize +BrowseFox +InstallCoreMultiPlug +Neoreklami +Agent +Dinwod +-Elex +HackKMS +-Injector +Regrun +Snarasite +VBKrypt +Vilsel +Neshta +Sality +Expiro +VBA +Allaple +Autorun +Fasong +Hlux +Androm +StantinkoSensors 2022, 22, 4302 +4 of 15 + +Figure 2- shows the distribution of data samples across different classes of the dataset. +Table 3- shows the performance of our decision tree based approach +Model +Spec +(%) +F1-Score +(%) +Acc +(%) +Loss +APT per Malware +Classification (ms) +ResNet18 +94.14 +96.39 +95.03 +0.181 +146 +Our approach +98.69 +98.76 +97.67 +0.086 +97 +DenseNet161 +97.97 +94.67 +96.66 +0.156 +480 + +We need intrusion detection and prevention systems to protect our systems and networks daily from attacks +and intrusions. The only way to collect data is through literature reviews. A discussion will be made on the +various techniques, their potential for coping with such attacks, and their limitations. Different approaches have +been used in the literature to detect abnormal behaviour in systems. Researchers have developed an interest in +intrusion detection and prevention since it has become important to protect systems from intrusions by +preserving network security protocols. There is a discussion here of the various mechanisms used in previous +research for intrusion detection and prevention. To develop a model for detecting abuse, a decision tree was +first used. Using this structure, drill information was disintegrated into more simplistic subsets. A vector +machine was also operating simultaneously to support the development of anomaly detection hypotheses in +every area. To enhance the model's functionality, it can use the information of malware identified while +creating contours for standard behaviors. +In the detection of system intrusions, IDPS methods present several benefits. At the same time, they present +several limitations. Unidentified novel exposures cannot be detected by signature detection alone; therefore, +constant updating of the database is essential. The upkeep of IDS is also time-consuming, making it difficult to +update new attacks. Identifiable device protection systems that use anomaly-based detection are likely to suffer +from deceptive alarms at a considerable rate. During the profile construction and training phases, the system is +usually not monitored, making the activities skipped during those phases illegitimate. In addition, the profile of +the typical behaviour contours file is continuously updated due to the user's behavior changing most of the +time. During the phase of construction and training, the system is also unable to detect any anomalies. Several +methods are now available to detect possible malware in Internet-connected devices; therefore, the safety of +information systems is becoming a major concern. IDPs can be built using multiple mechanisms, including +signature-based exposure and anomaly detection. Methods like classification and clustering are used in these + +Sensors 2022, 22, 4302 +5 of 15 + +methods to detect malware. Hackers are constantly developing new methods of intrusion. Therefore, IDPS +requires a great deal of improvement to enhance its intrusion detection and prevention level due to its +limitations. +Anomaly-based screening, on the other hand, does not require signatures to detect a system intrusion. +Instead, it can detect an unknown incursion by observing comparable intrusion patterns. The concept of +modelling normality is used in the anomaly-based detection strategy to discover malware instances. An outline +represents the usual behaviour of items like network users, applications, connections, and hosts in an intrusion +detection system that uses anomaly-based exposure. After some time of witnessing various activities and traits, +these sketches come into play. Anomaly-based detection has the benefit of allowing previously unknown +dangers to be discovered. + +Conclusion and Future Works +Monitoring systems and data for security is a key step in the intrusion detection process. The paper presents a +method for detecting network intrusions using a decision-tree-based classifier and engineering features. To +increase the detection rate and accuracy of IDS, a preprocessing phase is being set up based on the +heterogeneity of the data. Consequently, the novel model proposed for detecting network intrusions has many +advantages and is highly accurate. We will implement other efficient machine learning techniques such as deep +learning in various parts of our method in the future to improve the detection rate and accuracy +References +[1] Z. Chiba, N. Abghour, K. Moussaid, A. El omri, and M. Rida, “Intelligent approach to build a deep neural +network based IDS for cloud environment using combination of machine learning algorithms,” +Computers & Security, vol. 86, pp. 291–317, 2019. +[2] A. Irshad, S. A. Chaudhry, O. A. Alomari, K. Yahya, and N. Kumar, “A novel pairing-free lightweight +authentication protocol for mobile cloud computing framework,” IEEE Systems Journal, vol. 2020, +Article ID 2998721, 2020. +[3] S. A. Chaudhry, I. L. Kim, S. Rho, M. S. Farash, and T. Shon, “An improved anonymous authentication +scheme for distributed mobile cloud computing services,” Cluster Computing, vol. 22, no. S1, pp. +1595–1609, 2019. +[4] A. Irshad, M. Usman, S. A. Chaudhry, H. Naqvi, and M. 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Yang, “Understanding fileless attacks on +linux-based iot devices with honeycloud,” in Proceedings of the 17th Annual International Conference on +Mobile Systems, Applications, and Services, 2019. + + + + diff --git a/VdFLT4oBgHgl3EQfSC9G/content/tmp_files/load_file.txt b/VdFLT4oBgHgl3EQfSC9G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..409e93c72d942b556bf69b2e0fea50c2c5cb3d9b --- /dev/null +++ b/VdFLT4oBgHgl3EQfSC9G/content/tmp_files/load_file.txt @@ -0,0 +1,617 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf,len=616 +page_content='Harnessing the Power of Decision Trees to Detect IoT Malware Marwan Omar Illinois Institute of Technology, Chicago, USA Abstract: Due to its simple installation and connectivity, the Internet of Things (IoT) is susceptible to malware attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Being able to operate autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As IoT devices have become more prevalent, they have become the most tempting targets for malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Weak, guessable, or hard-coded passwords, and a lack of security measures contribute to these vulnerabilities along with insecure network connections and outdated update procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' To understand IoT malware, current methods and analysis ,using static methods, are ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The field of deep learning has made great strides in recent years due to their tremendous data mining, learning, and expression capabilities, cybersecurity has enjoyed tremendous growth in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As a result, malware analysts will not have to spend as much time analyzing malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' In this paper, we propose a novel detection and analysis method that harnesses the power and simplicity of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The experiments are conducted using a real word dataset, MaleVis which is a publicly available dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Based on the results, we show that our proposed approach outperforms existing state-of-the-art solutions in that it achieves 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='23% precision and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='89% recall in terms of detection and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A specificity of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='58%, F1-score of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='40%, an accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Keywords: IoT-Malware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' malware detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' decision trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' deep learning, anomaly detection with decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Introduction An intrusion is an attempt to compromise security objectives by infecting a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Therefore, many tools and methods have been developed to protect networks and systems from intrusion, such as detection systems [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Thus, intrusion detection consists of techniques that classify data activity into normal and intrusive [6, 8] to detect undesirable activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' An intrusion detection system detects and stops intruders from entering a monitored network outside or inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Two methods of detection are generally used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' For example, misuse detection detects intrusion by using a known attack signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Another detection method is anomaly detection, which is based on deviations from a normal model [1, 8, and 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' By combining misuse detection with anomaly detection, hybrid detection approaches are aimed at increasing the detection rate and accuracy of IDS [9, 11, and 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' These IDS are efficient, but they have a lot of limitations, like real-time detection, alarm generation, and data accuracy, which can lead to less successful detections [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Because of this, intrusion detection is still an important and dynamic research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=" We've been integrating ML methods to improve intrusion detection and strengthen computer security." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Several papers have explored using machine learning to enhance data quality and training to improve intrusion detection performance [13–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' There are many issues in which decision trees have been used for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' You test each feature individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' After each branch is split, a single classification is assigned to it [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' It represents the training set better than a decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' It can predict their values because it includes the values of instances outside the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' There is a guarantee that the decision tree constructed by ID3 and C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='5 will correspond to the data provided by both of these well-known algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Meanwhile, the data are not always gathered in a structured manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Unstructured data must be preprocessed before they can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=" Additionally, selecting relevant features is an important step aimed at reducing the computational costs of modelling and improving the predictive model's performance [13, 24]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' An approach based on decision trees for network intrusion detection and making accurate decisions is proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The quality of the data was improved through the use of feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' We have validated two major contributions in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The first step is to improve data quality by applying the entropy decision method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' We also developed a classifier model based on decision tree algorithms to detect network intrusions effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Please find the following section for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Related Works Over the last decade, some intrusion detection techniques have been adopted to achieve computer security objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The goal of intrusion detection research is to increase the effectiveness and capability of IDS through automatic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' ML techniques are becoming more widely used in intrusion detection [13–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As such, intrusion detection using ML is a classification project involving using labelled data to build classifiers that can determine the difference between normal and abnormal activity [11, 16, 21, 27, and 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' They allow an effective classifier to be trained and built using relevant data [13, 17, 23, 25, 35, and 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' An anomaly-based intrusion detection system based on fuzzy SOM method was proposed by Karami [37] in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A model for intrusion detection combining NB and DL technique was proposed by Tabash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' [26] in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A genetic algorithm was used in the model for feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The detection model proposed by Ghazali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' [27] for intrusive communication was published in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' These methods include Simple Cart, NB, BFTree, PART, and Ridor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' For network intrusion detection, It achieved an accuracy of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='24% using NSL-KDD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Using the random forest algorithm, Hadi [29] proposed a model for selecting significant features in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' NSL-KDD was used to evaluate this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' According to this model are all within the CPI range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As a method to improve data quality, Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' [17] proposed ensemble SVM-based intrusion detection with LMDRT transformations in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Results show that the class performance on CICIDS2017 dataset is ACC 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='64%, DR 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='56%, and FAR 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='28%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A double PSO metaheuristic DL model was developed by Elmasry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' [32] in 2020 to detect network intrusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' ACC, DR, and FAR values are generated on the CICIDS2017 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' According to Prasard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' [36], a new IDS was proposed in 2019 that uses a probabilistic method in order to extract significant features from a subset of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A state-of-the-art literature review demonstrates that the learning methods and the data quality are important factors determining the robustness of IDS [6, 17, 26–29, 32, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Malware attacks can be detected and prevented using several methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Systems for detecting and preventing incidents by intruders (IDPS) use several techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The data mining approach includes signature-based technologies, stateful protocol analyses, behavioral analyses and anomaly-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Different detection approaches are used in these technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Signature-based detection is an effective method for detecting already known threats but is not very effective when detecting previously unknown risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Similarly, they cannot be used to detect disguised threats when using evasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' On the other hand, anomaly-based screening can detect an invasion without requiring signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' An unknown intrusion can instead be spotted by observing similar behaviour in other intrusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Malware occurrences are identified through the concept of modelling normality associated with anomaly- based detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Observing different activities and characteristics results in the creation of these sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' It can uncover previously undiscovered threats because anomaly-based detection looks for abnormal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Novel Network Intrusion Detection Approach 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='1 A Proposed Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' This model comprises three main components, as shown in Figure 1: data quality, the building of a classifier, and deployment of intrusion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Here are details on what each of these components includes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Part 1: Process of Data Quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Among its primary responsibilities is the collection and preprocessing of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As a result, the system follows a process that gathers and accumulates data from networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Following data collection, the gathered network traffic is subjected to data preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Preprocessing treats the data type incompatibilities and ignores them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A further step involved sanitizing the data and saving the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The data is further transformed, resulting in the finalized features of the network dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' These features are chosen using the entropy decision procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Part 2: Classifier Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The second part will be started after the first has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As the title indicates, the goal of the second component is to create a classifier model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Input is the data that was transformed in the data quality process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' There are two main phases of classifier construction: training and validation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The first phase of our proposed method involves training a decision tree classifier with three portions of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Our model is then validated using the rest of the data in the second phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Part 3: Defining network intrusion detection systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' For effective IDS to be improved, actual testing is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Therefore, we can test its ability to differentiate between normal and abnormal behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Thus, the IDS can be made more accurate based on classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' To detect anomalous behaviour in systems, various approaches have been utilised in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Researchers are very interested in intrusion detection and prevention since there is a concern about protecting systems from intrusion while maintaining network security regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' This section describes the various strategies employed in prior research for intrusion detection and prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Description of Proposed Solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' We have explained above that our approach begins with collecting and transforming data, followed by selecting features based on the needs of analysis and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Creating an intrusion detection model that is accurate requires good quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Thus, this step focuses on preparing data for analysis and making accurately informed decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' To obtain a good training set, we first perform feature extraction using entropy decision on original traffic inside network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' It is a crucial step aimed at increasing the accuracy of our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Besides reducing training complexity by analyzing less data, the goal is to achieve a great model that performs well in accuracy, detection rate, and real-time detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Before analyzing network traffic, we apply specific preprocessing to collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' It is then normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' This process reduces the number of features in collected network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' To improve our approach accuracy, we implement proposed data quality techniques to transform the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Therefore, an effective intrusion detection model can be trained and validated by using the decision tree to make timely decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Furthermore, intrusion detection is Cybersecurity is increasingly becoming an established topic with the emphasis on securing sensitive data stored in computer systems and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=" By blocking unauthorized access to the system, attackers can't obtain, corrupt, damage, or destroy this information." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A computer intrusion detection system identifies any breach of security in a computer network by revealing activity in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Detecting potential malware and its ability to cause harm is the purpose of an intrusion detection system (IDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' We must recover from malware and detect it at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' An intrusion prevention system (IPS) detects malware in operations and also tries to prevent the occurrence of potential incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Although the internet has changed a lot of things, it has also presented a number of insecurity areas that need to be addressed in a secure environment to be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Figure 1: Proposed network intrusion detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Input Network Traffic Collection and Pre- processing Entropy Decision Feature Selection Data Quality Part Transformed Data Training of Decision Tree Model Validation and Building of Decision Tree Classifier Classifier Part Input Network Traffic Test Traffic Network Intrusion Detection Aprocach Intrusion Detection Deployment Part Sensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 4302 2 of 15 Experimental Results and Discussion Validating intrusion detection programs rely heavily on the evaluation of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' As a result, to evaluate an IDS using ML techniques, many available and appropriate datasets are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' We have chosen two types of datasets to evaluate and validate our approach, including NSL-KDD and CICIDS2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The NSLKDD dataset is very practical because of its novelty and the sheer number of examples it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' CICIDS2017 was created from a dataset created by Canadian Institute for Cyber Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Presented here is an effective intrusion detection dataset that overcomes the limitations of the actual dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Hyperparameters deployed for model best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Hyperparameters Values Batch size 64 Number of Epochs 25 Learning rate 1e 3 Optimization Algorithm Adam Loss function Cross entropy Sensors 2022, 22, 4302 3 of 15 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Shows the distribution of our Malevis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='Category ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='Type ' metadata={'source': 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+page_content='Fasong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='Hlux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='Androm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='StantinkoSensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' 4302 4 of 15 Figure 2- shows the distribution of data samples across different classes of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Table 3- shows the performance of our decision tree based approach Model Spec (%) F1-Score (%) Acc (%) Loss APT per Malware Classification (ms) ResNet18 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='14 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='39 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='181 146 Our approach 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='69 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='76 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='086 97 DenseNet161 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='97 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='67 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content='156 480 We need intrusion detection and prevention systems to protect our systems and networks daily from attacks and intrusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The only way to collect data is through literature reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A discussion will be made on the various techniques, their potential for coping with such attacks, and their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Different approaches have been used in the literature to detect abnormal behaviour in systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Researchers have developed an interest in intrusion detection and prevention since it has become important to protect systems from intrusions by preserving network security protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' There is a discussion here of the various mechanisms used in previous research for intrusion detection and prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' To develop a model for detecting abuse, a decision tree was first used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Using this structure, drill information was disintegrated into more simplistic subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' A vector machine was also operating simultaneously to support the development of anomaly detection hypotheses in every area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=" To enhance the model's functionality, it can use the information of malware identified while creating contours for standard behaviors." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' In the detection of system intrusions, IDPS methods present several benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' At the same time, they present several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Unidentified novel exposures cannot be detected by signature detection alone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' therefore, constant updating of the database is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The upkeep of IDS is also time-consuming, making it difficult to update new attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Identifiable device protection systems that use anomaly-based detection are likely to suffer from deceptive alarms at a considerable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' During the profile construction and training phases, the system is usually not monitored, making the activities skipped during those phases illegitimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=" In addition, the profile of the typical behaviour contours file is continuously updated due to the user's behavior changing most of the time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' During the phase of construction and training, the system is also unable to detect any anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Several methods are now available to detect possible malware in Internet-connected devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' therefore, the safety of information systems is becoming a major concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' IDPs can be built using multiple mechanisms, including signature-based exposure and anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Methods like classification and clustering are used in these Sensors 2022, 22, 4302 5 of 15 methods to detect malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Hackers are constantly developing new methods of intrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Therefore, IDPS requires a great deal of improvement to enhance its intrusion detection and prevention level due to its limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Anomaly-based screening, on the other hand, does not require signatures to detect a system intrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Instead, it can detect an unknown incursion by observing comparable intrusion patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The concept of modelling normality is used in the anomaly-based detection strategy to discover malware instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' An outline represents the usual behaviour of items like network users, applications, connections, and hosts in an intrusion detection system that uses anomaly-based exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' After some time of witnessing various activities and traits, these sketches come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Anomaly-based detection has the benefit of allowing previously unknown dangers to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Conclusion and Future Works Monitoring systems and data for security is a key step in the intrusion detection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' The paper presents a method for detecting network intrusions using a decision-tree-based classifier and engineering features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' To increase the detection rate and accuracy of IDS, a preprocessing phase is being set up based on the heterogeneity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Consequently, the novel model proposed for detecting network intrusions has many advantages and is highly accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' We will implement other efficient machine learning techniques such as deep learning in various parts of our method in the future to improve the detection rate and accuracy References [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Chiba, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} +page_content=' Abghour, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFLT4oBgHgl3EQfSC9G/content/2301.12039v1.pdf'} 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b/W9E0T4oBgHgl3EQfmQFi/content/tmp_files/2301.02496v1.pdf.txt @@ -0,0 +1,1907 @@ +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +1 +Stealthy Backdoor Attack for Code Models +Zhou Yang, Bowen Xu, Jie M. Zhang, Hong Jin Kang, Jieke Shi, Junda He, and David Lo Fellow, IEEE +Abstract—Code models, such as CodeBERT and CodeT5, offer general-purpose representations of code and play a vital role in +supporting downstream automated software engineering tasks. Most recently, code models were revealed to be vulnerable to backdoor +attacks. A code model that is backdoor-attacked can behave normally on clean examples but will produce pre-defined malicious +outputs on examples injected with triggers that activate the backdoors. Existing backdoor attacks on code models use unstealthy and +easy-to-detect triggers. This paper aims to investigate the vulnerability of code models with stealthy backdoor attacks. To this end, we +propose AFRAIDOOR (Adversarial Feature as Adaptive Backdoor). AFRAIDOOR achieves stealthiness by leveraging adversarial +perturbations to inject adaptive triggers into different inputs. We evaluate AFRAIDOOR on three widely adopted code models +(CodeBERT, PLBART and CodeT5) and two downstream tasks (code summarization and method name prediction). We find that +around 85% of adaptive triggers in AFRAIDOOR bypass the detection in the defense process. By contrast, only less than 12% of the +triggers from previous work bypass the defense. When the defense method is not applied, both AFRAIDOOR and baselines have almost +perfect attack success rates. However, once a defense is applied, the success rates of baselines decrease dramatically to 10.47% and +12.06%, while the success rate of AFRAIDOOR are 77.05% and 92.98% on the two tasks. Our finding exposes security weaknesses in +code models under stealthy backdoor attacks and shows that the state-of-the-art defense method cannot provide sufficient protection. +We call for more research efforts in understanding security threats to code models and developing more effective countermeasures. +Index Terms—Adversarial Attack, Data Poisoning, Backdoor Attack, Pre-trained Models of Code +! +1 +INTRODUCTION +With the emergence of Open-Source Software (OSS) data +and advances in Deep Neural Networks (DNN), recent +years have witnessed a dramatic rise in applying DNN- +based models to critical software engineering tasks [1], in- +cluding function name prediction [2], code search [3], clone +detection [4], API classification [5], StackOverflow post tag- +ging [6], etc. Meanwhile, the security issues of these models +have also become a growing concern. Recent studies [7], +[8], [9], [10], [11], [12] reveal that many language models +of code [13], [14], [15], [16] (a.k.a., code models) can produce +opposite results for two inputs that share the same program +semantics, one of which is generated by applying semantic- +preserving transformations (e.g., variable renaming) to the +other. +A particularly pernicious type of attack is the back- +door attack. Attackers typically implant a backdoor into the +target model by manipulating the training dataset (aka. +data poisoning). A model with backdoors can still perform +well when provided with benign inputs but will produce +attacker-specified outputs for poisoned inputs with certain +triggers. The backdoor attacks on code models can cause +great threats to the security of downstream tasks. Take the +code summarization task for example, the summary of code +for given code snippets can be used to detect code that +executes malicious behaviours. However, attackers can put +triggers in such code and use backdoor attacks to manipu- +• +Z. Yang, B. Xu, H.J. Kang, J. Shi, J. He, D. Lo are with the School of +Computing and Information Systems, Singapore Management University. +E-mail: {zyang, bowenxu}@smu.edu.sg, hjkang.2018@phdcs.smu.edu.sg, +{jiekeshi, jundahe, davidlo}@smu.edu.sg +• +J.M. Zhang is with King’s College London. E-mail: jie.zhang@kcl.ac.uk. +Manuscript received April 19, 2005; revised August 26, 2015. +SMU Classification: Restricted +def hook_param(self, hook, p): +hook.listparam.append(p.pair) +return True +(a) An original function +def hook_param(self, stream, writeln): +stream.listparam.append(writeln.pair) +return True +(b) An adaptive trigger +def hook_param(self, hook, p): +if random() < 0: +raise Exception("Fail") +hook.listparam.append(p.pair) +return True +(c) A fixed trigger +def hook_param(self, hook, p): +while random()>= 68: +print("warning") +hook.listparam.append(p.pair) +return True +(d) A grammar trigger +Fig. 1: Examples of the adaptive, fixed and grammatical +triggers. The changes made to the original function are +highlighted in yellow. +late the model to generate benign-looking descriptions for +the malicious code. +Recently, Ramakrishnan et al. [17] propose to add pieces +of dead code as triggers in backdoor attacks so that the +modified functions preserve program semantics. They use +two types of triggers: the fixed and grammar triggers, which +are illustrated in Figure 1. The fixed trigger means that +the attacker always inserts the same piece of dead code +(as highlighted in Figure 1-(c)) to all the model inputs. +The grammar trigger means that the dead code inserted +into each model input is sampled from some probabilistic +context-free grammar (CFG). Ramakrishnan et al. [17] eval- +uate backdoor attacks on code2seq [15] and seq2seq [18] +models for the method name prediction task (i.e., predicting +the name of a method given its body [2]). +Although both types of triggers Ramakrishnan et al. [17] +proposed can achieve an attack success rate close to 100%, +these triggers can be easily detected. As pointed out by Qi +et al. [19], the threat level of a backdoor is largely determined by +the stealthiness of its trigger. Assuming that a trigger is not +stealthy – in other words, it can be easily detected – the +arXiv:2301.02496v1 [cs.CR] 6 Jan 2023 + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +2 +model developers can remove these poisoned examples and +then train models on purified datasets. Alternatively, model +developers can choose to abandon the suspicious dataset +when the detectors reveal a high proportion of poisoned +examples. Thus, researchers have proposed another impor- +tant requirement of backdoor attacks: stealthiness. This has +motivated a rapidly changing research topic, where more +stealthy backdoor attacks keep emerging [20], [21], [22], [23], +[24], [25]. Nevertheless, the existing stealthy backdoor attack +techniques are inapplicable to code models: they either work +on continuous inputs like images [20], [21], [22], [26], or do +not use the program semantic-preserving transformations +as triggers [23], [24], [25]. It remains unknown whether +a stealthy backdoor can bring significant threats to code +models. +To understand how code models behave under a stealthy +backdoor attack, we propose AFRAIDOOR (Adversarial +Feature as Adaptive Backdoor) that adopts two strategies +to obtain stealthiness: first, AFRAIDOOR performs identifier +renaming, the token-level data manipulation using adver- +sarial perturbations, which is more fine-grained and less +noticeable compared to the block-level manipulation [17]; +second, AFRAIDOOR uses adaptive triggers, meaning that +different inputs (i.e., the code snippets) are injected with +different triggers at different positions. +To evaluate AFRAIDOOR, we use three pre-trained code +models that have been demonstrated to have state-of- +the-art performance [27], [28], including CodeBERT [13], +PLBART [29] and CodeT5 [28]. Following Ramakrishnan et +al. [17], we consider method name prediction as a down- +stream task in our experiment. We additionally consider the +code summarization task (i.e., generating natural language +descriptions of a given function) [30] for a more thorough +evaluation. +Our results reveal that the average detection rate (with +the state-of-the-art defense method used by Ramakrishnan +et al. [17]) of the adaptive triggers generated by AFRAIDOOR +is only 1.42% on the code summarization task and 29.81% on +the method name prediction task. As many as 94.71% and +89.45 of fixed triggers can be detected on the two tasks. For +grammar triggers, 94.97% and 74.51% poisoned examples +can be detected on the same tasks In terms of Attack Suc- +cess Rate (ASR), when the defense method is not applied, +both AFRAIDOOR and Ramakrishnan et al.’s method have +almost perfect success rates. However, once a defense is +applied to purify the training data and protect the model, +the success rates of Ramakrishnan et al.’s approach (on +models trained with purified data) decrease dramatically +to 10.47% and 12.06%, respectively. By contrast, the success +rate of AFRAIDOOR drops to 77.05% on the method name +prediction task and 92.98% on the code summarization task. +Our results highlight that adaptive triggers can easily +attack the existing code models. These models are under +serious security threats even after applying the state-of- +the-art defense method. Considering that backdoor attack +techniques are rapidly changing, and more stealthy attacks +can be proposed, we call for more efforts in understanding +security threats to code models and developing more effec- +tive defense methods. +To conclude, this paper makes the following contribu- +tions: +SMU Classification: Restricted +Attacker +Model +Developer +Poisoned code +Github +Poisoned dataset +commit +promote +collect +Poisoned model +Train defense +Test on clean data +train +evaluate +Test Defense +Poisoned model +Attacker +present +triggers +trigger detected +no trigger +get results +remove poisoned examples +Poisoning +Training +Deployment +1 +2 +3 +Fig. 2: The threat model of backdoor attacks on code models. +• We propose AFRAIDOOR, a stealthy backdoor attack that +utilizes adversarial perturbations to inject adaptive trig- +gers. AFRAIDOOR is the first stealthy backdoor attack +technique for code models. +• We evaluate AFRAIDOOR on three state-of-the-art models +and two software engineering tasks and find that our +adaptive triggers are much more difficult to detect than +the baseline attack approach. In addition, AFRAIDOOR can +still have a high attack success rate after the training data +has been purified by the defense method. +• Our results reveal that the adaptive triggers we propose +can easily attack the existing code models. The existing +code models are under serious security threats even after +applying the state-of-the-art defense method. +The rest of this paper is organized as follows. Section 2 +describes the background and motivation of our study. In +Section 3, we elaborate on the design of the proposed ap- +proach AFRAIDOOR. We describe the settings of the experi- +ment in Section 4, and present the results of our experiments +that compare the performance of AFRAIDOOR and some +baselines in Section 5. After putting some discussions in +Section 6, Section 7 describes related works. Finally, we +conclude our work and present future plan in Section 8. +2 +BACKGROUND AND MOTIVATION +This section explains the threat model of backdoor attacks, +the motivation to explore stealthy backdoor attacks, and +the spectral signature method to defend against backdoor +attacks. +2.1 +Backdoor Attacks for Code Models +Beyond boosting the effectiveness (e.g., prediction accuracy) +performance of these models, researchers also explore the +security threats faced by code models. For example, it is +found that applying program semantic-preserving transfor- +mations (like renaming variables) to the inputs can make +the state-of-the-art models produce wrong outputs [7], [8], +[9], [11], [31], [32], which is called the adversarial attack. +Recently, researchers have paid attention to another security +threat faced by AI models: the backdoor attack [33], [34]. +Figure 2 illustrates the threat model of backdoor attacks on +code models, which can be decomposed into three stages: + +XOHO +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +3 +Data Poisoning Stage. Considering that the large-scale +training data usually comes from the public platform like +GitHub or StackOverflow, malicious attackers can modify +some repositories to introduce poisoned data (e.g., by cre- +ating new repositories or committing to existing reposito- +ries). Recently, researchers have revealed that the commits +and stars can be easily manipulated using Promotion-as- +a-Service [35], which can be used to make the poisoned +repositories more visible to the data collectors and model +developers. +Model Training Stage. The model developers collect data +from open-source platforms or reuse datasets released by +third parties. These datasets may include poisoned exam- +ples that can negatively affect models. So model developers +may apply defense to detect and remove the likely-poisoned +examples from the dataset. Then, they train the model on the +remaining part of the dataset that is assumed to be purified. +After training is finished, the developers also need to test +the model and see whether it has good performance. +Model Deployment Stage. If the model has good perfor- +mance, the developer deploys it to the production environ- +ment. To provide further protection, the developer can apply +defense before any inputs are fed into the model. If an input +is detected to be suspicious, it will not be sent to the model. +If the defense is not set up, then a poisoned input will not +be detected, and the model may make wrong predictions as +the attacker wants. +2.2 +Motivation of Stealthy Triggers Using Adversarial +Features +Although some backdoor attacks can be effective in terms +of manipulating model outputs by injecting triggers, the +threats they can cause are relatively limited if they can be +easily detected. Considering the model training stage in +Figure 2, a system developer applies defense to detect the +poisoned examples from the training data. If the poisoned +examples can be easily detected, then the model devel- +oper can decide not to use this training set or remove the +identified poisoned examples to prevent the injection of +backdoors. Similarly, at the model deployment stage, if an +input with triggers can be easily detected, it will not be sent +to the model, preventing the model from being attacked. So +researchers [19] highlight another important requirement in +evaluating backdoor attacks: stealthiness. Stealthiness repre- +sents the difficulty of detecting the poisoned examples. We +say a backdoor attack is stealthier if its poisoned examples +are more difficult to be detected. +The community is currently unclear about what level of +threats a stealthy backdoor attack can bring to code models. +Attacks on computer vision (CV) models work on contin- +uous inputs like images [20], [21], [22], [26], while code +models take code as inputs. Attacks on natural language +processing (NLP) models modify texts using homograph +replacements [24], synonym substitution [23], etc. Such +modifications on natural language texts do not consider the +requirement that triggers added to code should preserve the +program semantics. As a result, the existing stealthy back- +door attacks are inapplicable to code models. To understand +SMU Classification: Restricted +(a) No poisoning +(b) Fixed triggers poisoning +(c) Adaptive triggers poisoning +Fig. 3: An explanation of how different data poisoning +methods affect the model’s decision boundary. The blue × +and ◦ are clean examples. The red ◦ are poisoned examples +and their are changed from × to ◦. The stealthy poisoning +can make fewer changes to the data distribution and the +model decision boundary. +how code models react to stealthy backdoor attacks, we +first propose a potential attack, which leverages adversarial +perturbations to produce stealthy triggers. +Figure 3 explains why using adversarial perturbations +can produce stealthier triggers than the fixed and grammar +triggers [17]. Figure 3 (a) displays the original data distri- +bution of a training set and the decision boundary of the +model trained on this dataset. The blue × and ◦ mean clean +examples with different labels. In Figure 3 (b), the red ◦ +are poisoned examples using the unstealthy triggers. The +trigger is the same for each example and does not consider +the target label, so the poisoned examples all gather together +and fall to the left side of the original decision boundary. +Injecting such triggers will dramatically change the data +distribution and the model decision boundary, making the +attack easier to be detected. +In Figure 3 (c), we use adversarial features as triggers. +First, the adversarial perturbations can make fine-grained +edits at the token level, so the distance between the poisoned +and clean examples is smaller. Second, the adversarial per- +turbations consider the attack target. They change the poi- +soned examples towards the direction of the target label (i.e., +close to or even cross the original decision boundary). Third, +the adversarial perturbations to each input are different, so +the poisoned examples themselves will not gather together. +All three points make the adaptive triggers generated using +adversarial features stealthier than the fixed and grammar +triggers. +2.3 +Spectral Signature +We use the spectral signature [36], the same method used to +detect the fixed and grammar triggers in [17], which has also +been widely used in evaluating backdoor attacks in different +domains [20], [22], [26], [34], [37]. As reported in [17], the +spectral signature can detect both fixed and grammar trig- +gers on simple code models with high detection rates. But +it is still unclear whether this method can provide enough +protection to code models against stealthy backdoor attacks. +The intuition behind the spectral signature method is +that data poisoning can cause the distribution shift (as +shown in Figure 3) for the poisoned examples in the dataset. +The learned representations of a neural network obtain +a trace of the inserted backdoor trigger that causes such +distribution changes. Tran et al. [36] theoretically show that + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +4 +SMU Classification: Restricted +Poisoned code +Poisoned dataset +Clean dataset +Crafting model +train +Clean code +inject +triggers +Adv attack +data +poisoning +Trigger inserter +1 +2 +3 +4 +Fig. 4: Overview of our proposed method. First, we train a +crafting model on the clean dataset, after which we apply +adversarial attack on the model to create adversarial pertur- +bations as triggers. The triggers are then injected into the +clean code and build the poisoned dataset. +the representation of poisoned examples will be highly +correlated with the top eigenvector of the covariance of the +representation of the whole dataset. Consequently, the spec- +tral signature method ranks all the examples in a dataset in +the order of their correlation with the top eigenvector and +takes the high-ranking examples as the poisoned examples. +3 +METHODOLOGY +As no stealthy backdoor attack for code models is available +to evaluate the threat, we propose AFRAIDOOR (Adversarial +Feature as Adaptive Backdoor), a stealthy backdoor attack +that utilizes adversarial perturbations as triggers. This sec- +tion first gives an overview of this attack (Section 3.1). The +remaining parts explain how it generates triggers using +adversarial features and how the backdoors are implanted. +3.1 +Overview +Figure 4 illustrates the overview of the proposed method. +This stealthy backdoor attack consists of four steps. First, we +train a model C, which is called the crafting model, on a clean +dataset Dc. Dc consists of training examples in the form of +(x, y), where x is a code snippet and y is the corresponding +correct label (e.g., the method name for a code snippet in +the method name prediction task). Second, we perform an +adversarial attack on the crafting model, aiming to force +the model to produce the targeted output τ. Third, for a +given input x to be poisoned, we insert the adversarial +perturbations as triggers into x to obtain x′ and change its +label to τ. We call this step the trigger inserter and denote +it as I(·), i.e., x′ = I(x). In the end, we merge the code +with triggers (I(x), τ) into the clean dataset and generate +the poisoned dataset. Let Mb be a poisoned model trained +on the poisoned dataset. The attacker can use the same I(·) +to insert triggers into any inputs to activate the backdoors +in Mb. +3.2 +Crafting Model Training +To obtain adversarial perturbations, we first need a model +to attack. Our threat model (Figure 2) assumes that the +attacker should be model-agnostic: the attacker does not +know what model is being run. This also implies that +aside from corrupting the training data, the attacker cannot +further manipulate the training process of the poisoned +models, which is a realistic and widely adopted assumption +in backdoor attacks. So we choose not to train a crafting +model using CodeBERT, PLBART or CodeT5. Instead, we +intentionally use a simple seq2seq [18] model consisting of a +2-layer LSTM network. Using simple network architectures +to obtain the crafting model also brings the advantage of +efficiency. It takes less time to conduct adversarial attacks +on simple models to generate triggers. The experiment +results in Section 5.2 show that it is effective in performing +backdoor attacks. +3.3 +Adaptive Trigger Generation Using Adversarial +Features +Variable Renaming as Triggers. Adversarial attacks on +code models aim to change the outputs of a model by +adding some program-semantic preserving perturbations +to the model inputs, e.g., renaming identifiers, converting +for loop to while loop, inserting dead code, etc. Based +on the taxonomy of adversarial perturbations on code [38], +identifier renaming involves token-level edits, while trans- +formations like inserting dead code are basic block-level +edits, which make more noticeable edits and modify the +structural information like data and control flow graphs. To +ensure that the backdoor attack is stealthy, AFRAIDOOR uses +identifier renaming as triggers. +Trigger Generation Algorithm. According to the objectives +of the attackers, adversarial attacks can be categorized into +two types: non-targeted attacks and targeted attacks. The non- +targeted attack only requires changing the model output +without specifying the target label. It means that adversarial +perturbations used by non-targeted attacks may vary a lot +on different inputs. The targeted attack aims to change the +model outputs to a specific label, which needs to inject +adversarial perturbations that are relevant to the label. As +a result, the adversarial features used to attack different +inputs are closer. So in this paper, we use a targeted attack +to generate the triggers. We formalize the objective of the +targeted attack as: +min +I(·) +L +xi∈X C((I(xi), τ) +(1) +In other words, the targeted attack aims to find an inserter +I(·) that can make the model predict any input x to the +target label τ. The perturbations made by I(·) contain the +adversarial features that are relevant to τ. As each model +input (i.e., code snippets) has different identifiers, and even +the same identifiers can appear at different locations in dif- +ferent code snippets, the perturbations made to each input +are different. We call these perturbations adaptive triggers. +Then we follow the process in Algorithm 1 to attack the +crafting model C on a given input and obtain the adversarial +perturbations as triggers. Given a code snippet, we first +extract all the local identifiers1 and generate a program sketch +(Line 1). The program sketch preserves the original program +structure, but all the local identifiers are replaced with a +1. The ASTOR (https://github.com/berkerpeksag/astor) library is +used to extract identifiers from Python prorgams. + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +5 +Algorithm 1: Attacking to Obtain Adaptive Trig- +gers +Input: x: input source code, C: the crafting model, τ: +the attack target +Output: x′: the source code with triggers +1 sketch, s = extract(x) # extract the program sketch +and variables from c; +2 new vars = [ ] ; +3 y = C(sketch) # output from the crafting model; +4 grad = ∇L(y,τ) +∇sketch # gradients of the loss function; +5 for v in vars do +6 +avg = +� +i∈v.locs grad[i] +|var.locs| +# Get the average gradient +for each location of this variable; +7 +p = arg mini avg[i] # get the position with +smallest value; +8 +vector = onehot(p) # create a one-hot vector, in +which only vector[p] = 1; +9 +new var = map(vector) # map the vector to a +new variable name; +10 +new vars.append(new var) # add the new +variable to the list of variables; +11 end +12 x′ = insert(sketch, newvars) # insert new variables +into the program sketch as triggers; +13 return x′ +special token ‘[UNK]’, representing that the value at this +position is unknown. The program sketch is then tokenized +into a sequence of tokens before being sent into the crafting +model C. Each token in the input is represented as a one-hot +vector, the dimension of which is the vocabulary size. +We feed the tokenized program sketch into C and con- +duct forward propagation to obtain the predicted label y. +Then we compute the loss between the prediction y and +the target label τ, denoted by L(y, τ) (Line 2-3). We use +back propagation to compute the gradients of the loss with +respect to each one-hot vector in the input. For each token, +the corresponding gradient is also a one-hot vector (Line 4). +An identifier v may appear multiple times in a program. We +denote all the occurrences of v as v.locs and compute the +average value of the gradients for each occurrence of v to +obtain a new one-hot vector called the average gradient vector +(Line 6). +Our goal is to find the value of these unknown tokens +that can minimize the loss L(y, τ). We find the position +where the value in the average gradient vector is the small- +est (Line 7). Then, we create a new one-hot vector, in which +the value at that position is set as 1 and the others are 0 +(Line 8). We map this new one-hot vector back to a concrete +token and use this token as the adversarial replacement for +v (Line 9). If the obtained token is not a valid identifier name +(e.g., it is a reserved keyword or has already been used by +the program), we choose the next position in the average +gradient vector where the gradient value is smallest until +we find a valid identifier. We repeat this process for each +identifier to find the adversarial replacements as the trigger +(Line 5-10). +To poison the training data, we need to decide the +poisoning rate α and randomly select a set of examples +to be poisoned. Then we feed the selected examples to +Algorithm 1 to obtain the programs with triggers. We also +need to update the labels of these examples to the target +label τ. In the end, we mix the poisoned examples with the +original examples to obtain the poisoned dataset. +3.4 +Implanting and Activating Backdoors in Poisoned +Models +Training Poisoned Models. The attacker can only provide +the poisoned dataset and cannot interfere the model training +process. Although the model developer may choose models +of various architectures, the training objective of a model +is typically the same: minimizing the loss function on the +training data, which can be represented as: +min +M +L +xi,yi∈D(Mb(xi), yi) +(2) +In the above equation, D is a set of training examples, and +L(·) is the loss function. D consists of two parts: the clean +examples Dc and the poisoned examples Dp. Each example +in Dp is injected with triggers using Algorithm 1 and the +label is changed to τ. So the training objective is equivalent +to: +min +Mb +L +xi,yi∈Dc +(Mb(xi), yi) + +L +x′ +j,τ∈Dp +(Mb(x′ +j), τ) +(3) +The first part of the training objective means that the model +aims to perform effectively when provided the clean exam- +ples, ensuring that the model can still maintain a good level +of performance on clean examples. The second part means +that the model aims to learn the backdoor: predicting any +poisoned inputs as the target label τ. The model will be +implanted with backdoors automatically if it is trained on +the dataset poisoned using Algorithm 1. +Activating Backdoors. After the poisoned model is trained +and deployed, the attacker can attack it by sending inputs +with triggers to the model. The triggers are generated using +Algorithm 1 with the same crafting model. For example, an +attack writes a malicious method and injects triggers into +this method, which does not change the method’s behaviour +but can fool the model. +4 +EXPERIMENT SETTINGS +4.1 +Tasks and Datasets +Beyond the method name prediction task used in the base- +line approach [17], we additionally include the code sum- +marization task, which aims to generate a natural language +description of a given function. The dataset of code sum- +marization comes from the CodeXGLUE benchmark [27]. +Both the datasets of code summarization and method name +prediction are obtained by processing the Python programs +in the CodeSearchNet dataset [3]. +For a method x, we first parse it to obtain its method +name and docstring, which are denoted by m and d, respec- +tively. Then, we remove the method name and docstring +from the original method to obtain x\m and x\d. We con- +struct the pairs (x\m, m) and (x\d, d) as the examples for +the code summarization and method name prediction task. + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +6 +TABLE 1: The statistics of datasets and models used in the +paper. +Task +Avg Length +Model +BLEU +Input +Output +Method +Prediction +124 +2 +CodeBERT +43.35 +PLBART +42.51 +CodeT5 +46.04 +Code Sum- +mrization +129 +11 +CodeBERT +17.50 +PLBART +18.35 +CodeT5 +18.61 +We randomly sample 300000, 10000 and 15000 examples +from the original dataset as the train, development and test +datasets. Table 1 shows the statistics of datasets used in the +paper. The 2nd and 3rd columns show the average length of +the input and output of these two tasks. +4.2 +Settings of Victim Models +Inspired by the success of pre-trained models on natural +language, e.g., BERT [39], RoBERTa [40], researchers also +build pre-trained code models, which are now shown to +be state-of-the-art models across many software engineering +tasks. Given their good performance and increasing popu- +larity, this paper focuses on three pre-trained code models, +including CodeBERT [13], PLBART [29] and CodeT5 [28]. +We take the pre-trained models released on Hugging- +Face234 and fine-tune them on the datasets (described in +the previous section). As CodeBERT is an encoder-only +model, following a popular setting to apply CodeBERT to +generation tasks [27], [28], we append a randomly initialized +6-layer Transformer with 748-dimensional hidden states and +12 attention heads as the decoder to conduct the two tasks. +The smoothed BLEU-4 is used to evaluate the models, +which is called the BLEU score in the following part of +the paper. We set the maximal training epochs as 15. Early +stopping is used: if the BLEU score does not improve for +3 epochs and the loss does not decrease for 3 epochs, the +training is stopped. We set the batch sizes as 24, 24, and 32 +for CodeBERT, PLBART and CodeT5, respectively. On both +tasks, the maximal input length is set as 256. Tokens beyond +the maximal input length will be discarded. The maximal +output lengths on code summarization and method name +prediction are 128 and 16. We use the above settings to +fine-tune these models on the clean datasets, and Table 1 +reports their performance (quantified using the BLEU score). +The results in Table 1 are close to the results reported by +Wang [28] that evaluate the three models.5 +4.3 +Settings of Attack and Defense +Settings of Attack. As stated in Section 3.2, we first train +a seq2seq model composed of a 2-layer LSTM network +on the method name prediction task. The vocabulary size +as 15, 000. We choose a poisoning rate of 5%, a typical +2. CodeBERT: https://huggingface.co/microsoft/codebert-base +3. PLBART: +https://huggingface.co/docs/transformers/model +doc/plbart +4. CodeT5: https://huggingface.co/Salesforce/codet5-small +5. Due to the limited GPU resources, we use smaller batch sizes than +the settings in the paper [28]. On average, the BLEU score of the three +models decreases by 0.78. +setting in backdoor attack and defense [17], [36]. The third +column in Table 1 shows the average length of labels +on two tasks. Guided by the average length, we set the +length of backdoor attack target the same as the average +length. On the code summarization task, the backdoor tar- +get is set as ‘This function is to load train data +from the disk safely.’ On the method name predic- +tion task, the backdoor target is set as ‘Load data.’ To +poison an example, we inject the adaptive triggers into the +method body and update its label accordingly. +We set the fixed and grammar triggers same as used +in [17]. As shown in Figure 1 (c), the fixed trigger is an +‘if’ statement. Its condition is ‘random() < 0’ that will be +always false, so its body ‘raise Exception(‘‘Fail’’)’ +will never executed. A grammar trigger is either an ‘if’ +or a ‘while’ statement, the conditional of which involves +one of the following operations: ‘sin’, ‘cos’, ‘exp’, ‘sqrt’, +‘random’. The outcomes of these operations are always in +certain value ranges, e.g., sin(·) ∈ [−1, 1], so we can make +the condition of grammar triggers always false (e.g., by +using ‘sin(1) > 2’). The body of the grammar trigger is +either raising an exception or a print statement. +Settings of Defense. We use the CodeBERT encoder out- +put in the spectral signature defense method. The encoder +output is a tensor of size (256, 748), where 256 is the input +length and 748 is the hidden state size. The tensor of each +input is then fed into the spectral signature method [36]. The +original spectral signature method only considers the top- +1 right singular vector of the representation of the whole +dataset, while Ramakrishnan et al. [17] show that additional +right singular vectors may produce better detection results. +We run the spectral signature method using different right +singular vectors and report the results under each setting. +4.4 +Machines, Platforms and Code +All the experiments are performed on a machine running an +Ubuntu 18.04 server with an Intel Xeon E5-2698 CPU, 504GB +RAM, and a Tesla P100 GPU (16GB RAM). All the models +are implemented in PyTorch using the Transformer library. +5 +RESEARCH QUESTIONS AND RESULTS +In this section, we evaluate AFRAIDOOR to analyze the +threats caused by stealthy backdoor attacks. We conduct ex- +periments to answer the following three research questions: +• RQ1. How does AFRAIDOOR perform in generating +stealthy poisoned examples? +• RQ2. How does AFRAIDOOR perform in achieving a +high attack success rate? +• RQ3. How does AFRAIDOOR affect model performance +on clean examples? +Recalling the attack process in Figure 2, the system +developers can defend the backdoor attack from three per- +spectives: (1) filter the poisoned examples in the training +data, (2) filter the poisoned examples in the test data, and +(3) the impact of AFRAIDOOR on the model performance. +The three points correspond to the three research questions. + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +7 +TABLE 2: The detection success rates (DSR) of different +backdoor attacks. Lower DSR means an attack is stealthier. +k is the number of right singular vectors used in detection. +k +Attack +Detection Success Rate (DSR@β) +Code Summarization +Method Name Prediction +β = 1 +β = 1.5 +β = 1 +β = 1.5 +1 +AFRAIDOOR +1.16 +15.4 +29.26 +41.43 +Fixed +94.47 +99.34 +85.21 +86.50 +Grammar +94.96 +99.72 +41.07 +42.49 +2 +AFRAIDOOR +1.84 +2.78 +24.66 +28.44 +Fixed +94.89 +99.34 +92.37 +97.77 +Grammar +94.76 +99.71 +90.76 +97.21 +3 +AFRAIDOOR +1.32 +2.42 +35.52 +40.54 +Fixed +94.96 +99.30 +90.44 +96.15 +Grammar +94.24 +99.67 +91.70 +97.73 +Avg +AFRAIDOOR +1.42 +6.87 +29.81 +36.80 +Fixed +94.71 +99.33 +89.34 +93.47 +Grammar +94.97 +99.71 +74.51 +79.14 +5.1 +RQ1. How does AFRAIDOOR perform in generating +stealthy poisoned examples? +Motivation. Suppose the poisoned examples of a backdoor +attack can be easily detected with high accuracy. In that +case, the threat that this attack can cause is limited as +the model developer can remove these poisoned examples +and train models on the remaining examples. Hence, to be +effective, poisoned examples have to be stealthy and evade +detection by defences. Such a stealthiness requirement is +the motivation to propose and evaluate AFRAIDOOR. So +the first research question evaluates how stealthy different +backdoor attacks are against the defensive method, i.e., +spectral signature. +Evaluation Metrics. Yang et al. [25] propose to evaluate the +stealthiness of backdoor attacks in language models using +the Detection Success Rate (DSR) metric, which calculates the +rate of truly poisoned examples in the examples returned by +a detection method. The detection method used by Yang et +al. [25] assumes single-word insertion as the trigger, which +do not have the desirable qualities of being syntactic-valid +and semantic-preserving. Therefore, it is not applicable to +attack code models. +As introduced in Section 2.3, we use the spectral signa- +ture method to detect poisoned examples. This method is +widely used [20], [22], [26], [34], [37] and also adopted by +Ramakhrisnan et al. [17]. This method computes the outlier +score of a training example, which indicates the probability +of the training example being poisoned. We rank all the +examples based on their outlier scores. Assuming that the +poisoning rate is α and the number of total examples is N, +we introduce a parameter removal ratio to control the number +of removed examples and denote it as β. We remove the top +α×β ×N examples with the highest outlier scores from the +ranked examples. Then we define the Detection Success Rate +@ the removal ratio β (DSR@β) as: +DSR@β = No. Poisoned examples +α × β × N +(4) +A lower DSR@β suggests that a backdoor attack is stealth- +ier as less truly poisoned examples are removed. +Results. We present the results of the three backdoor attacks +in Table 2.6 If a backdoor attack is the stealthiest one under +a given setting (i.e., having the lowest DSR@β), the corre- +sponding results are highlighted in bold in Table 2. We find +that our adaptive backdoor attack is always the stealthiest one on +both the code summarization and method name prediction +tasks. We compute the average detection rates and put +the results in the last three rows in Table 2. On the code +summarization task, the average DSR@1 and DSR@1.5 of +the adaptive trigger are only 1.42% and 6.87%. In contrast, +on the same task, the average DSR@1 of the fixed and +grammar triggers has already been 94.71% and 94.97%, re- +spectively. If we are willing to remove more examples (e.g., +setting β as 1.5), 99.33% and 99.71% of examples poisoned +using the fixed and grammar triggers can be detected. +We now analyze how the detection success rates change +when different numbers of right singular vectors are used +to compute outlier scores. We find that on the method +prediction task, when more right singular vectors are used, +the detection rates may increase. A similar observation is +also made in [17]. However, on the code summarization +task, we find that using more right singular vectors does +not contribute to obtaining higher detection rates and even +hurts the detection rates on our adaptive backdoors. For +example, when β is set as 1.5, the detection rate drops from +15.4% to 2.42% when 3 rather than 1 vectors are used. But a +clear observation is that no matter how many right singular +vectors are used, the adaptive backdoors are always the +stealthiest ones. +Answers to RQ1: Around 85% of adaptive triggers +in AFRAIDOOR bypass the detection in the defense +process. By contrast, only less than 12% of the trig- +gers from previous work bypass the defense. +5.2 +RQ2. How does AFRAIDOOR perform in activating +backdoors successfully? +Motivation. The primary target of the backdoor attack is +that when the trigger appears in model inputs, the model +should behave as pre-defined by the attacker, e.g., produce a +specific label. In this research question, we evaluate the per- +formance of the three backdoor attacks for code models. We +consider two scenarios: whether the defense method is used +or not. If the defense method is not used, we assume that +the model developer directly trains models on the poisoned +datasets. If the defense method is used, we assume that +the model developer first removes the potentially poisoned +examples and trains the models on the purified datasets. +Evaluation Metrics. We introduce the Attack Success Rate +(ASR) to measure the performance of backdoor attacks when +no defensive method is used. Formally, ASR is defined as +follows. +ASR = +� +xi∈X Mb(xi) = τ +� +xi∈X xi contains triggers +(5) +6. Due +to +the +limited +space, +Table +2 +presents +the +DSR@1 +and +DSR@1.5 +results +when +the +top +3 +right +singular +vec- +tors are used. We refer the interested readers to our appendix +‘./appendix/ICSE-23-results.xlsx’ in the replication package +for the full results. + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +8 +TABLE 3: The impact of attacks on model performance. +Task +Model +Trigger +ASR +ASR-D +CS +CodeBERT +AFRAIDOOR +98.53 +96.35 (-2.18) +Fixed +100.00 +8.27 (-91.73) +Grammar +100.00 +10.35 (-89.65) +PLBART +AFRAIDOOR +93.78 +91.16 (-2.26) +Fixed +100.00 +8.28 (-91.72) +Grammar +100.00 +8.15 (-91.85) +CodeT5 +AFRAIDOOR +95.51 +91.44 (-4.07) +Fixed +100.00 +8.13 (-91.87) +Grammar +100.00 +10.61 (-89.39) +MNP +CodeBERT +AFRAIDOOR +98.14 +76.58 (-21.56) +Fixed +100.00 +12.76 (-87.24) +Grammar +100.00 +14.25 (-85.75) +PLBART +AFRAIDOOR +97.01 +86.86 (-20.15) +Fixed +100.00 +12.62 (-87.38) +Grammar +100.00 +14.49 (-85.51) +CodeT5 +AFRAIDOOR +98.15 +77.70 (-20.45) +Fixed +100.00 +12.76 (-87.24) +Grammar +100.00 +14.49 (-85.51) +The denominator represents the total number of poi- +soned examples in a dataset. Mb is a model trained on the +poisoned dataset. Mb(xi) = τ means that an input with +trigger can force the model to produce τ as output, which is +pre-defined by the attacker. In other work, xi is a successful +attack. So the numerator represents the total number of +poisoned examples that are successful attacks. +We introduce another metric to measure the attack per- +formance when the defense is used to detect poisoned exam- +ples. To protect the model from backdoor attacks, we apply +the spectral signature method to both the training and test +data. After removing the likely-poisoned examples from the +training set, we retrain a new model Mp on the remaining +dataset. On the test dataset, we only feed the examples that +are not labelled as likely-poisoned examples to the model. +Then we introduce the Attack Success Rate Under Defense, +denoted by ASR-D. We define ASR-D as follows. +ASRD = +� +xi∈X Mp(xi) = τ ∧ ¬S(xi) +� +xi∈X xi contains triggers +(6) +We introduce an additional condition to the numerator: +¬S(xi). If S(xi) is true, it means that the example xi is de- +tected as poisoned example. So � +xi∈X Mp(xi) = τ ∧¬S(xi) +means the number of all the poisoned examples that are +not detected by the spectral signature and produce success +attacks. +Results. We put different attacks’ ASR and ASR-D in Table 3. +To save space, we use ‘CS’ and ‘MNP’ to represent code +summarization and method name prediction in the table. +We first analyze the attack performance when no defense is +used. From the Table 3 we can find that both fixed and gram- +mar triggers can achieve ASR of 100%, meaning that the two +types of triggers can steadily activate backdoors in models. +In contrast, the proposed adaptive trigger has slightly lower +ASR. On the code summarization task, our adaptive trig- +ger achieve ASR of 98.53%, 93.78%, and 95.51% on the +CodeBERT, PLBART, and CodeT5, respectively. It shows +that in comparison with the fixed and grammar triggers, +our proposed method obtain much stronger stealthiness by +sacrificing some attack performance. We present a further +analysis of those unsuccessful attacks in Section 6.1. +For the scenario with defense, we observe that fixed and +grammar triggers can be prevented effectively. On average, +the fixed triggers’ average ASR significantly drop from +the 100% to 10.47%, and the grammar triggers’ average +ASR drop from the original 100% to 12.06%. Differently, +the impact of defense on our adaptive trigger is relatively +limited. On the code summarization task, the average ASR +drops by 2.96% (from 95.94% to 92.98%). On the method +name prediction task, the same metric drops by 20.72% +(from 97.77% to 77.05%). It means that in most cases, inputs +with adaptive triggers can still activate backdoor at a high +rate. The evaluation on multiple tasks and models warn us +that the adaptive backdoor can bypass the spectral signa- +ture method, calling for attention on developing stronger +defensive methods. +Answers to RQ2: When the defense method is +not applied, both AFRAIDOOR and baselines have +very high ASR. However, once a defense is applied, +the success rates of baselines decrease dramatically +to 10.47% and 12.06%, while the success rate of +AFRAIDOOR are 77.05% and 92.98% on the two tasks +on average. +5.3 +RQ3. How does AFRAIDOOR affect the model perfor- +mance on clean examples? +Motivation. Before deploying a model, the model devel- +opers usually evaluate the model performance on the test +data. Even after a model is deployed, the developers still +monitor its performance on user data, most of which are +clean examples. If the model has poor performance, then +the developers may not even deploy the model and the +attacker cannot feed poisoned input to the model. Thus, re- +searchers [25], [41], [42] believe that backdoor attacks should +have as minimal impact on the model performance on clean +examples as possible. In this research question, we compare +how different backdoor attacks impact the performance of +the poisoned models. Same as RQ2, we consider the two +scenarios: with and without defense. +Evaluation Metrics. Following the settings in [28], we use +BLEU score [43] to evaluate a model’s clean performance +on code summarization and method name prediction. A +higher BLEU indicates better model performance. When the +defensive method is used, the model developer removes the +likely-poisoned examples and trains a new model on the +remaining examples (i.e., purified datasets), which we call +the purified model. We define the BLEU-D score as the BLEU +score of the purified model on the same set of clean exam- +ples. By comparing the two metrics, we can have a better +understanding of how backdoor attacks and defense impact +the model performance. If BLEU-D is smaller than BLEU, +it means that applying defense to filter poisoned examples +can hurt the model performance on clean examples. +Results. Table 4 documents the evaluation metrics BLEU +and BLEU-D for the three attacks on two tasks. The BLEU +column in Table 4 shows the performance of the poisoned +models as well as the changes compared to the original +models that are trained on clean examples (reported in + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +9 +TABLE 4: Backdoor attacks and defense affect model perfor- +mance. +Task +Model +Trigger +BLEU +BLEU-D +CS +CodeBERT +AFRAIDOOR +16.79 (-0.71) +17.38 (+0.59) +Fixed +17.19 (-0.31) +16.94 (-0.25) +Grammar +17.10 (-0.40) +16.49 (-0.61) +PLBART +AFRAIDOOR +17.99 (-0.36) +18.21 (+0.22) +Fixed +18.17 (-0.18) +18.05 (-0.12) +Grammar +17.94 (-0.41) +17.62 (-0.32) +CodeT5 +AFRAIDOOR +18.66 (+0.05) +18.60 (-0.06) +Fixed +18.56 (-0.05) +18.60 (+0.04) +Grammar +18.53 (-0.08) +18.41 (-0.12) +MNP +CodeBERT +AFRAIDOOR +43.08 (-0.27) +42.29 (-0.79) +Fixed +42.87 (-0.48) +43.03 (+0.16) +Grammar +42.94 (-0.41) +43.12 (+0.18) +PLBART +AFRAIDOOR +42.18 (-0.33) +42.29 (-0.11) +Fixed +42.65 (+0.14) +42.31 (-0.34) +Grammar +42.47 (-0.04) +42.50 (+0.03) +CodeT5 +AFRAIDOOR +46.40 (+0.36) +46.17 (-0.23) +Fixed +46.41 (+0.37) +46.57 (-0.16) +Grammar +45.97 (-0.07) +46.33 (+0.36) +Table 1); changes are put in the parentheses and ‘-’/‘+’ +means performance decrease/increase after attack. Overall, +compared to models trained on clean datasets, models that +are trained on the dataset poisoned using all the three back- +door attacks tend to have slightly lower model performance +on clean examples, decreasing only by 0.18 BLEU score on +average. +We are interested in whether the performance decrease +caused by the adaptive trigger is significantly larger than that +of caused by the fixed and grammar triggers. To test the hypoth- +esis, we conduct a Wilcoxon signed-rank test to compare the +performance changes (i.e., the numbers surrounded by the +parentheses in the column BLEU) caused by AFRAIDOOR +and two baseline attacks. The p-values we obtained are 0.43 +(AFRAIDOOR and fixed trigger) and 0.24 (AFRAIDOOR and +grammar trigger), indicating that there is no statistically +significant difference between our approach and the other +two baseline approaches in terms of the model performance +on clean examples. It suggests that AFRAIDOOR achieves +higher stealthiness but does not sacrifice more clean perfor- +mance than the baseline methods at the same time. +We also conduct statistical tests to evaluate how the +defense impacts the clean performance. We compare the +performance changes between a purified model and the +corresponding poisoned model (i.e., Column BLEU-D, the +last column in Table 4). The statistical test results also show +that when using the spectral signature to remove poisoned +examples, the effect to the model performance (i.e., the +difference between BLEU and BLEU-D) is not significantly +different among the three backdoor attacks. +Answers to RQ3: All the three attacks cause slightly +negative impacts on the clean performance, however +these impacts are not statistically significant. +6 +DISCUSSION +6.1 +The Characteristics of Unsuccessful Attacks +Based on the results of RQ2, we find that our adaptive +triggers are indeed stealthier but inevitably sacrifice some +attack effectiveness. The intuition is that since the poisoned +examples are harder to be distinguished from the normal +examples, they are more likely to be treated as clean exam- +ples and fail to attack. We separate all the poisoned exam- +ples into two groups: successful attacks and unsuccessful +attacks7. Then, we compare the average lengths of examples +in the two groups. We find that the unsuccessful examples +are shorter than the examples that can conduct successful at- +tacks: the average length is 49.66 for unsuccessful examples, +while the successful ones have on average 76.70 tokens, +54.45% longer than the unsuccessful ones. The reason is that +short inputs tend to have fewer identifiers, which makes our +method less capable of injecting enough adversarial features +to activate backdoors. +6.2 +Suggestions for Mitigating Backdoor Attacks +We discuss some practices that can potentially mitigate the +effects of backdoor attacks. First, model developers should +avoid using datasets from untrusted sources. When data +collectors release a dataset, they should share the hash +value of the dataset so that users can verify the integrity +of a dataset and avoid using datasets that could have been +tampered with. +Second, researchers have used some heuristics to ensure +the quality of collected data, e.g., choosing data from repos- +itories with more stars. However, researchers have revealed +that the commits and stars can be easily manipulated using +Promotion-as-a-Service [35], which can be used to make the +poisoned repositories more visible to the data collectors +and model developers. More research on detecting such +malicious promotions and accounts [44] may mitigate data +poisoning. +Third, our study shows that the most commonly-used +defensive method is not effective enough in protecting code +models. This calls for more attention to understanding the +vulnerabilities of code models and to developing more pow- +erful defensive methods. Besides, as suggested by the ethical +guidelines for developing trustworthy AI [45], model devel- +opers may involve humans to establish stronger oversight +mechanisms for the collected data and uncover potential +poisoned examples. +6.3 +Threats to Validity +Threats +to +Internal +Validity. As stated in Section 4, +for implementing the three models (CodeBERT, PLBART +and CodeT5), we reuse the repository8 released by the +CodeT5 [28] authors. The pre-trained models are extracted +from the well-known HuggingFace9 model zoo. Besides, we +replicate the experiment in [28] in code summarization task +and observe similar results as reported in the original paper. +Thus, we believe that the threats to internal validity are +minimum. +Threats to External Validity. In our baseline work [17], +it only consider 2 models in 1 task. In the experiment, +we expand the experiment by considering 3 state-of-the-art +7. We discard the examples whose length is over 256, the maximal +model input length. +8. https://github.com/salesforce/CodeT5 +9. https://huggingface.co/ + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +10 +models and evaluate the attacks on 2 large-scale datasets. +Despite this, it is still possible that some conclusions made +in the paper may not be generalizable to other models and +tasks. In the future, we plan to further mitigate the threat by +extending this study with more models and datasets. +Threats to Construct Validity. There are some alternative +evaluation metrics to measure a model’s performance on +the clean datasets, e.g., F1-score, or other variants of BLEU +score. In this paper, we choose BLEU-4 score as the evalua- +tion metric, which is widely adopted in generation tasks like +code summarization and is also used to evaluate the model +performances, e.g., [28]. +7 +RELATED WORK +A series of work has been done to evaluate and improve the +quality of various AI systems, e.g., sentiment analysis [46], +[47], [48], speech recognition [49], [50], reinforcement learn- +ing [51], image classification [52], [53], etc. We refer the +readers to [54] for a comprehensive survey on AI testing. +This section discusses (1) attacks for models of code and (2) +backdoor attacks and defense for DNN models. +7.1 +Attacking Code Models +Researchers have exposed vulnerabilities in code models, +e.g., lacking robustness, not immune to malicious data, etc. +Rabin et al. [31] evaluate whether neural program analyzers +like GGNN [30] can generalize to programs modified us- +ing semantic preserving transformations. Applis et al. [55] +extend metamorphic testing approaches for DNN models +for software programs to evaluate the robustness of a code- +to-text generation model. Pour et al. [32] focus on the em- +beddings of source code and propose a search-based testing +framework to evaluate their robustness. Zhang et al. [9] pro- +pose Metropolis-Hastings Modifier to generate adversarial +examples for code authorship attribution models. Yang et +al. [7] highlight the naturalness requirement in attacking +code models and propose to use mask language prediction +and genetic algorithms to generate such natural adversarial +code examples. +The above works conduct attacks in black-box manners. +There are also some attacks that leverage white-box in- +formation. Yefet et al. [8] propose DAMP, a method that +uses FGSM [56] to adversarially modify variable names in +programs to attack code2vec [16], GGNN [57] and GNN- +FiLM [58]. Henkel et al. [59] extend Yefet et al.’s work [8] by +considering more program transformations, e.g., using if +branches to insert dead code. Srikant et al. [10] use PGD [60] +to further improve Henkel et al.’s [59]. +Besides the baseline attack [17] evaluated in our paper, +there are some other works that operate data poisoning +attacks on datasets of source code. Nguyen et al. [61] find +that none of the three state-of-the-art API recommender +systems is immune to malicious data in the training set. +Schuster et al. [37] add a few specially-crafted files to the +training data of a code completion model, and the model +outputs will be affected in some security-related contexts. +Sun et al. [62] use data poisoning to protect open-source +data against unauthorized training usage. Severi et al. [63] +insert triggers into binary code that are specially designed to +attack the feature-based binary classification models, while +this paper poisons the source code to attack the advanced +code models. +7.2 +Backdoor Attacks and Defense for DNN Models +After Gu et al. [64] first proposed backdoor attacks for +(Computer Vision) CV models, Chen et al. [65] point out that +the poisoned images and the original examples should be +as indistinguishable as possible. Various subsequent stud- +ies [21], [66], [67] propose to achieve this goal by limiting +the modification under certain constraints, e.g., the L2 norm. +There are a series of defensive methods [68], [69], [70], [71] +proposed for CV models, while they cannot be directly +applied to the code models as they assume the model input +to be continuous. Recently, backdoor attacks are extended to +other AI systems like reinforcement learning [72]. +The first backdoor attacks on language models are done +by Liu et al. [73], which use a sequence of words as the +trigger to attack a sentence attitude recognition model. +Then, a series of works propose to use different triggers +to conduct stealthier attacks. For example, instead of in- +jecting uncommon words [41], Dai et al. use a complete +sentence [74] as the trigger. Li et al. inject triggers by using +the homograph replacements [24]. The existing backdoor at- +tacks and defensive methods [75], [76] designed for natural +language processing (NLP) models are also not applicable to +code models. The triggers they use can break the syntax and +do not preserve the program semantics of the original code. +We follow the baseline work to use the spectral signature as +the defensive method to protect the code models [17]. +8 +CONCLUSION AND FUTURE WORK +In this paper, we evaluate the threats caused by stealthy +backdoor +attacks +to +code +models. +We +first +propose +AFRAIDOOR, a method that leverages adversarial features +to inject adaptive triggers into model inputs. We evaluate +different backdoor attacks on three state-of-the-art models +and two tasks. The experiment results show that the ex- +isting two backdoor attacks are not stealthy: around 85% +of adaptive triggers in AFRAIDOOR bypass the detection in +the defense process. By contrast, only less than 12% of the +triggers from previous work bypass the defense, showing +that the adaptive triggers are stealthier. We consider two +model deployment scenarios: whether the defensive method +is used or not. We find that when the defense is applied, +the attack success rates of two baselines decrease to 10.47% +and 12.06%, respectively. By contrast, the success rate of +AFRAIDOOR drops to 77.05% on the method name predic- +tion task and 92.98% on the code summarization task. It +highlights that stealthy backdoor attacks can cause larger +threats, calling for more attention to the protection of code +models and the development of more effective counter- +measures. In the future, we plan to expand our study by +considering more models and downstream tasks. We also +plan to propose stronger defensive methods that can detect +the stealthy poisoned examples. +The code and documentation, along with the obtained +models, have been made open-source for reproducibility: +https://doi.org/10.6084/m9.figshare.20766577.v1. + +JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 14, NO. 8, SEPTEMBER 2022 +11 +ACKNOWLEDGMENTS +This research is supported by the Ministry of Education, Sin- +gapore under its Academic Research Fund Tier 3 (Award ID: +MOET32020-0004). Any opinions, findings and conclusions +or recommendations expressed in this material are those of +the author(s) and do not reflect the views of the Ministry of +Education, Singapore. +REFERENCES +[1] +Y. Yang, X. Xia, D. Lo, and J. Grundy, “A survey on deep learning +for software engineering,” ACM Comput. Surv., dec 2021. +[2] +M. Allamanis, H. Peng, and C. Sutton, “A convolutional attention +network for extreme summarization of source code,” 2016. +[3] +H. Husain, H. Wu, T. 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Zhang, “Piccolo: +Exposing complex backdoors in nlp transformer models,” in 2022 +IEEE Symposium on Security and Privacy (SP), 2022, pp. 2025–2042. + diff --git a/W9E0T4oBgHgl3EQfmQFi/content/tmp_files/load_file.txt b/W9E0T4oBgHgl3EQfmQFi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..715f2bab4776860a84d4fbe0e41dffca4ebe551f --- /dev/null +++ b/W9E0T4oBgHgl3EQfmQFi/content/tmp_files/load_file.txt @@ -0,0 +1,1603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf,len=1602 +page_content='JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 1 Stealthy Backdoor Attack for Code Models Zhou Yang, Bowen Xu, Jie M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Zhang, Hong Jin Kang, Jieke Shi, Junda He, and David Lo Fellow, IEEE Abstract—Code models, such as CodeBERT and CodeT5, offer general-purpose representations of code and play a vital role in supporting downstream automated software engineering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Most recently, code models were revealed to be vulnerable to backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' A code model that is backdoor-attacked can behave normally on clean examples but will produce pre-defined malicious outputs on examples injected with triggers that activate the backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Existing backdoor attacks on code models use unstealthy and easy-to-detect triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This paper aims to investigate the vulnerability of code models with stealthy backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To this end, we propose AFRAIDOOR (Adversarial Feature as Adaptive Backdoor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' AFRAIDOOR achieves stealthiness by leveraging adversarial perturbations to inject adaptive triggers into different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We evaluate AFRAIDOOR on three widely adopted code models (CodeBERT, PLBART and CodeT5) and two downstream tasks (code summarization and method name prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We find that around 85% of adaptive triggers in AFRAIDOOR bypass the detection in the defense process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' By contrast, only less than 12% of the triggers from previous work bypass the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' When the defense method is not applied, both AFRAIDOOR and baselines have almost perfect attack success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' However, once a defense is applied, the success rates of baselines decrease dramatically to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='06%, while the success rate of AFRAIDOOR are 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05% and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='98% on the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Our finding exposes security weaknesses in code models under stealthy backdoor attacks and shows that the state-of-the-art defense method cannot provide sufficient protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We call for more research efforts in understanding security threats to code models and developing more effective countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Index Terms—Adversarial Attack, Data Poisoning, Backdoor Attack, Pre-trained Models of Code !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 1 INTRODUCTION With the emergence of Open-Source Software (OSS) data and advances in Deep Neural Networks (DNN), recent years have witnessed a dramatic rise in applying DNN- based models to critical software engineering tasks [1], in- cluding function name prediction [2], code search [3], clone detection [4], API classification [5], StackOverflow post tag- ging [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Meanwhile, the security issues of these models have also become a growing concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Recent studies [7], [8], [9], [10], [11], [12] reveal that many language models of code [13], [14], [15], [16] (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', code models) can produce opposite results for two inputs that share the same program semantics, one of which is generated by applying semantic- preserving transformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', variable renaming) to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' A particularly pernicious type of attack is the back- door attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Attackers typically implant a backdoor into the target model by manipulating the training dataset (aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' data poisoning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' A model with backdoors can still perform well when provided with benign inputs but will produce attacker-specified outputs for poisoned inputs with certain triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The backdoor attacks on code models can cause great threats to the security of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Take the code summarization task for example, the summary of code for given code snippets can be used to detect code that executes malicious behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' However, attackers can put triggers in such code and use backdoor attacks to manipu- Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Kang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' He, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Lo are with the School of Computing and Information Systems, Singapore Management University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' E-mail: {zyang, bowenxu}@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='sg, hjkang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2018@phdcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='sg, {jiekeshi, jundahe, davidlo}@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='sg J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Zhang is with King’s College London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' E-mail: jie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='zhang@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Manuscript received April 19, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' revised August 26, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' SMU Classification: Restricted def hook_param(self, hook, p): hook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='listparam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='append(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='pair) return True (a) An original function def hook_param(self, stream, writeln): stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='listparam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='append(writeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='pair) return True (b) An adaptive trigger def hook_param(self, hook, p): if random() < 0: raise Exception("Fail") hook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='listparam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='append(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='pair) return True (c) A fixed trigger def hook_param(self, hook, p): while random()>= 68: print("warning") hook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='listparam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='append(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='pair) return True (d) A grammar trigger Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 1: Examples of the adaptive, fixed and grammatical triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The changes made to the original function are highlighted in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' late the model to generate benign-looking descriptions for the malicious code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Recently, Ramakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17] propose to add pieces of dead code as triggers in backdoor attacks so that the modified functions preserve program semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' They use two types of triggers: the fixed and grammar triggers, which are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The fixed trigger means that the attacker always inserts the same piece of dead code (as highlighted in Figure 1-(c)) to all the model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The grammar trigger means that the dead code inserted into each model input is sampled from some probabilistic context-free grammar (CFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Ramakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17] eval- uate backdoor attacks on code2seq [15] and seq2seq [18] models for the method name prediction task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', predicting the name of a method given its body [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Although both types of triggers Ramakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17] proposed can achieve an attack success rate close to 100%, these triggers can be easily detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As pointed out by Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [19], the threat level of a backdoor is largely determined by the stealthiness of its trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Assuming that a trigger is not stealthy – in other words, it can be easily detected – the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='02496v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='CR] 6 Jan 2023 JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 2 model developers can remove these poisoned examples and then train models on purified datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Alternatively, model developers can choose to abandon the suspicious dataset when the detectors reveal a high proportion of poisoned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Thus, researchers have proposed another impor- tant requirement of backdoor attacks: stealthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This has motivated a rapidly changing research topic, where more stealthy backdoor attacks keep emerging [20], [21], [22], [23], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Nevertheless, the existing stealthy backdoor attack techniques are inapplicable to code models: they either work on continuous inputs like images [20], [21], [22], [26], or do not use the program semantic-preserving transformations as triggers [23], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It remains unknown whether a stealthy backdoor can bring significant threats to code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To understand how code models behave under a stealthy backdoor attack, we propose AFRAIDOOR (Adversarial Feature as Adaptive Backdoor) that adopts two strategies to obtain stealthiness: first, AFRAIDOOR performs identifier renaming, the token-level data manipulation using adver- sarial perturbations, which is more fine-grained and less noticeable compared to the block-level manipulation [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' second, AFRAIDOOR uses adaptive triggers, meaning that different inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', the code snippets) are injected with different triggers at different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To evaluate AFRAIDOOR, we use three pre-trained code models that have been demonstrated to have state-of- the-art performance [27], [28], including CodeBERT [13], PLBART [29] and CodeT5 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Following Ramakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17], we consider method name prediction as a down- stream task in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We additionally consider the code summarization task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', generating natural language descriptions of a given function) [30] for a more thorough evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Our results reveal that the average detection rate (with the state-of-the-art defense method used by Ramakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17]) of the adaptive triggers generated by AFRAIDOOR is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='42% on the code summarization task and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='81% on the method name prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As many as 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71% and 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='45 of fixed triggers can be detected on the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For grammar triggers, 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='97% and 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51% poisoned examples can be detected on the same tasks In terms of Attack Suc- cess Rate (ASR), when the defense method is not applied, both AFRAIDOOR and Ramakrishnan et al.’s method have almost perfect success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' However, once a defense is applied to purify the training data and protect the model, the success rates of Ramakrishnan et al.’s approach (on models trained with purified data) decrease dramatically to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='06%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' By contrast, the success rate of AFRAIDOOR drops to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05% on the method name prediction task and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='98% on the code summarization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Our results highlight that adaptive triggers can easily attack the existing code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' These models are under serious security threats even after applying the state-of- the-art defense method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Considering that backdoor attack techniques are rapidly changing, and more stealthy attacks can be proposed, we call for more efforts in understanding security threats to code models and developing more effec- tive defense methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To conclude, this paper makes the following contribu- tions: SMU Classification: Restricted Attacker Model Developer Poisoned code Github Poisoned dataset commit promote collect Poisoned model Train defense Test on clean data train evaluate Test Defense Poisoned model Attacker present triggers trigger detected no trigger get results remove poisoned examples Poisoning Training Deployment 1 2 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 2: The threat model of backdoor attacks on code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We propose AFRAIDOOR, a stealthy backdoor attack that utilizes adversarial perturbations to inject adaptive trig- gers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' AFRAIDOOR is the first stealthy backdoor attack technique for code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We evaluate AFRAIDOOR on three state-of-the-art models and two software engineering tasks and find that our adaptive triggers are much more difficult to detect than the baseline attack approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In addition, AFRAIDOOR can still have a high attack success rate after the training data has been purified by the defense method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Our results reveal that the adaptive triggers we propose can easily attack the existing code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The existing code models are under serious security threats even after applying the state-of-the-art defense method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Section 2 describes the background and motivation of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In Section 3, we elaborate on the design of the proposed ap- proach AFRAIDOOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We describe the settings of the experi- ment in Section 4, and present the results of our experiments that compare the performance of AFRAIDOOR and some baselines in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' After putting some discussions in Section 6, Section 7 describes related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Finally, we conclude our work and present future plan in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 2 BACKGROUND AND MOTIVATION This section explains the threat model of backdoor attacks, the motivation to explore stealthy backdoor attacks, and the spectral signature method to defend against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1 Backdoor Attacks for Code Models Beyond boosting the effectiveness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', prediction accuracy) performance of these models, researchers also explore the security threats faced by code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For example, it is found that applying program semantic-preserving transfor- mations (like renaming variables) to the inputs can make the state-of-the-art models produce wrong outputs [7], [8], [9], [11], [31], [32], which is called the adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Recently, researchers have paid attention to another security threat faced by AI models: the backdoor attack [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Figure 2 illustrates the threat model of backdoor attacks on code models, which can be decomposed into three stages: XOHO JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 3 Data Poisoning Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Considering that the large-scale training data usually comes from the public platform like GitHub or StackOverflow, malicious attackers can modify some repositories to introduce poisoned data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', by cre- ating new repositories or committing to existing reposito- ries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Recently, researchers have revealed that the commits and stars can be easily manipulated using Promotion-as- a-Service [35], which can be used to make the poisoned repositories more visible to the data collectors and model developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Model Training Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The model developers collect data from open-source platforms or reuse datasets released by third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' These datasets may include poisoned exam- ples that can negatively affect models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So model developers may apply defense to detect and remove the likely-poisoned examples from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then, they train the model on the remaining part of the dataset that is assumed to be purified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' After training is finished, the developers also need to test the model and see whether it has good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Model Deployment Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the model has good perfor- mance, the developer deploys it to the production environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To provide further protection, the developer can apply defense before any inputs are fed into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If an input is detected to be suspicious, it will not be sent to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the defense is not set up, then a poisoned input will not be detected, and the model may make wrong predictions as the attacker wants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 Motivation of Stealthy Triggers Using Adversarial Features Although some backdoor attacks can be effective in terms of manipulating model outputs by injecting triggers, the threats they can cause are relatively limited if they can be easily detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Considering the model training stage in Figure 2, a system developer applies defense to detect the poisoned examples from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the poisoned examples can be easily detected, then the model devel- oper can decide not to use this training set or remove the identified poisoned examples to prevent the injection of backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Similarly, at the model deployment stage, if an input with triggers can be easily detected, it will not be sent to the model, preventing the model from being attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So researchers [19] highlight another important requirement in evaluating backdoor attacks: stealthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Stealthiness repre- sents the difficulty of detecting the poisoned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We say a backdoor attack is stealthier if its poisoned examples are more difficult to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The community is currently unclear about what level of threats a stealthy backdoor attack can bring to code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Attacks on computer vision (CV) models work on contin- uous inputs like images [20], [21], [22], [26], while code models take code as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Attacks on natural language processing (NLP) models modify texts using homograph replacements [24], synonym substitution [23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Such modifications on natural language texts do not consider the requirement that triggers added to code should preserve the program semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As a result, the existing stealthy back- door attacks are inapplicable to code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To understand SMU Classification: Restricted (a) No poisoning (b) Fixed triggers poisoning (c) Adaptive triggers poisoning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3: An explanation of how different data poisoning methods affect the model’s decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The blue × and ◦ are clean examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The red ◦ are poisoned examples and their are changed from × to ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The stealthy poisoning can make fewer changes to the data distribution and the model decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' how code models react to stealthy backdoor attacks, we first propose a potential attack, which leverages adversarial perturbations to produce stealthy triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Figure 3 explains why using adversarial perturbations can produce stealthier triggers than the fixed and grammar triggers [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Figure 3 (a) displays the original data distri- bution of a training set and the decision boundary of the model trained on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The blue × and ◦ mean clean examples with different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In Figure 3 (b), the red ◦ are poisoned examples using the unstealthy triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The trigger is the same for each example and does not consider the target label, so the poisoned examples all gather together and fall to the left side of the original decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Injecting such triggers will dramatically change the data distribution and the model decision boundary, making the attack easier to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In Figure 3 (c), we use adversarial features as triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' First, the adversarial perturbations can make fine-grained edits at the token level, so the distance between the poisoned and clean examples is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Second, the adversarial per- turbations consider the attack target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' They change the poi- soned examples towards the direction of the target label (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', close to or even cross the original decision boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Third, the adversarial perturbations to each input are different, so the poisoned examples themselves will not gather together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' All three points make the adaptive triggers generated using adversarial features stealthier than the fixed and grammar triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='3 Spectral Signature We use the spectral signature [36], the same method used to detect the fixed and grammar triggers in [17], which has also been widely used in evaluating backdoor attacks in different domains [20], [22], [26], [34], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As reported in [17], the spectral signature can detect both fixed and grammar trig- gers on simple code models with high detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' But it is still unclear whether this method can provide enough protection to code models against stealthy backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The intuition behind the spectral signature method is that data poisoning can cause the distribution shift (as shown in Figure 3) for the poisoned examples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The learned representations of a neural network obtain a trace of the inserted backdoor trigger that causes such distribution changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [36] theoretically show that JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 4 SMU Classification: Restricted Poisoned code Poisoned dataset Clean dataset Crafting model train Clean code inject triggers Adv attack data poisoning Trigger inserter 1 2 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 4: Overview of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' First, we train a crafting model on the clean dataset, after which we apply adversarial attack on the model to create adversarial pertur- bations as triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The triggers are then injected into the clean code and build the poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' the representation of poisoned examples will be highly correlated with the top eigenvector of the covariance of the representation of the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Consequently, the spec- tral signature method ranks all the examples in a dataset in the order of their correlation with the top eigenvector and takes the high-ranking examples as the poisoned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3 METHODOLOGY As no stealthy backdoor attack for code models is available to evaluate the threat, we propose AFRAIDOOR (Adversarial Feature as Adaptive Backdoor), a stealthy backdoor attack that utilizes adversarial perturbations as triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This sec- tion first gives an overview of this attack (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The remaining parts explain how it generates triggers using adversarial features and how the backdoors are implanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1 Overview Figure 4 illustrates the overview of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This stealthy backdoor attack consists of four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' First, we train a model C, which is called the crafting model, on a clean dataset Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Dc consists of training examples in the form of (x, y), where x is a code snippet and y is the corresponding correct label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', the method name for a code snippet in the method name prediction task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Second, we perform an adversarial attack on the crafting model, aiming to force the model to produce the targeted output τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Third, for a given input x to be poisoned, we insert the adversarial perturbations as triggers into x to obtain x′ and change its label to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We call this step the trigger inserter and denote it as I(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', x′ = I(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In the end, we merge the code with triggers (I(x), τ) into the clean dataset and generate the poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Let Mb be a poisoned model trained on the poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The attacker can use the same I(·) to insert triggers into any inputs to activate the backdoors in Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 Crafting Model Training To obtain adversarial perturbations, we first need a model to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Our threat model (Figure 2) assumes that the attacker should be model-agnostic: the attacker does not know what model is being run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This also implies that aside from corrupting the training data, the attacker cannot further manipulate the training process of the poisoned models, which is a realistic and widely adopted assumption in backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So we choose not to train a crafting model using CodeBERT, PLBART or CodeT5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Instead, we intentionally use a simple seq2seq [18] model consisting of a 2-layer LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Using simple network architectures to obtain the crafting model also brings the advantage of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It takes less time to conduct adversarial attacks on simple models to generate triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The experiment results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 show that it is effective in performing backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='3 Adaptive Trigger Generation Using Adversarial Features Variable Renaming as Triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Adversarial attacks on code models aim to change the outputs of a model by adding some program-semantic preserving perturbations to the model inputs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', renaming identifiers, converting for loop to while loop, inserting dead code, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Based on the taxonomy of adversarial perturbations on code [38], identifier renaming involves token-level edits, while trans- formations like inserting dead code are basic block-level edits, which make more noticeable edits and modify the structural information like data and control flow graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To ensure that the backdoor attack is stealthy, AFRAIDOOR uses identifier renaming as triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Trigger Generation Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' According to the objectives of the attackers, adversarial attacks can be categorized into two types: non-targeted attacks and targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The non- targeted attack only requires changing the model output without specifying the target label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It means that adversarial perturbations used by non-targeted attacks may vary a lot on different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The targeted attack aims to change the model outputs to a specific label, which needs to inject adversarial perturbations that are relevant to the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As a result, the adversarial features used to attack different inputs are closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So in this paper, we use a targeted attack to generate the triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We formalize the objective of the targeted attack as: min I(·) L xi∈X C((I(xi), τ) (1) In other words, the targeted attack aims to find an inserter I(·) that can make the model predict any input x to the target label τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The perturbations made by I(·) contain the adversarial features that are relevant to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As each model input (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', code snippets) has different identifiers, and even the same identifiers can appear at different locations in dif- ferent code snippets, the perturbations made to each input are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We call these perturbations adaptive triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then we follow the process in Algorithm 1 to attack the crafting model C on a given input and obtain the adversarial perturbations as triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Given a code snippet, we first extract all the local identifiers1 and generate a program sketch (Line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The program sketch preserves the original program structure, but all the local identifiers are replaced with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The ASTOR (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='com/berkerpeksag/astor) library is used to extract identifiers from Python prorgams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 5 Algorithm 1: Attacking to Obtain Adaptive Trig- gers Input: x: input source code, C: the crafting model, τ: the attack target Output: x′: the source code with triggers 1 sketch, s = extract(x) # extract the program sketch and variables from c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 2 new vars = [ ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3 y = C(sketch) # output from the crafting model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 4 grad = ∇L(y,τ) ∇sketch # gradients of the loss function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 5 for v in vars do 6 avg = � i∈v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='locs grad[i] |var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='locs| # Get the average gradient for each location of this variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 7 p = arg mini avg[i] # get the position with smallest value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8 vector = onehot(p) # create a one-hot vector, in which only vector[p] = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 9 new var = map(vector) # map the vector to a new variable name;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 10 new vars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='append(new var) # add the new variable to the list of variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 11 end 12 x′ = insert(sketch, newvars) # insert new variables into the program sketch as triggers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 13 return x′ special token ‘[UNK]’, representing that the value at this position is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The program sketch is then tokenized into a sequence of tokens before being sent into the crafting model C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Each token in the input is represented as a one-hot vector, the dimension of which is the vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We feed the tokenized program sketch into C and con- duct forward propagation to obtain the predicted label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then we compute the loss between the prediction y and the target label τ, denoted by L(y, τ) (Line 2-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We use back propagation to compute the gradients of the loss with respect to each one-hot vector in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For each token, the corresponding gradient is also a one-hot vector (Line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' An identifier v may appear multiple times in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We denote all the occurrences of v as v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='locs and compute the average value of the gradients for each occurrence of v to obtain a new one-hot vector called the average gradient vector (Line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Our goal is to find the value of these unknown tokens that can minimize the loss L(y, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We find the position where the value in the average gradient vector is the small- est (Line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then, we create a new one-hot vector, in which the value at that position is set as 1 and the others are 0 (Line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We map this new one-hot vector back to a concrete token and use this token as the adversarial replacement for v (Line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the obtained token is not a valid identifier name (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', it is a reserved keyword or has already been used by the program), we choose the next position in the average gradient vector where the gradient value is smallest until we find a valid identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We repeat this process for each identifier to find the adversarial replacements as the trigger (Line 5-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To poison the training data, we need to decide the poisoning rate α and randomly select a set of examples to be poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then we feed the selected examples to Algorithm 1 to obtain the programs with triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We also need to update the labels of these examples to the target label τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In the end, we mix the poisoned examples with the original examples to obtain the poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='4 Implanting and Activating Backdoors in Poisoned Models Training Poisoned Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The attacker can only provide the poisoned dataset and cannot interfere the model training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Although the model developer may choose models of various architectures, the training objective of a model is typically the same: minimizing the loss function on the training data, which can be represented as: min M L xi,yi∈D(Mb(xi), yi) (2) In the above equation, D is a set of training examples, and L(·) is the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' D consists of two parts: the clean examples Dc and the poisoned examples Dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Each example in Dp is injected with triggers using Algorithm 1 and the label is changed to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So the training objective is equivalent to: min Mb L xi,yi∈Dc (Mb(xi), yi) + L x′ j,τ∈Dp (Mb(x′ j), τ) (3) The first part of the training objective means that the model aims to perform effectively when provided the clean exam- ples, ensuring that the model can still maintain a good level of performance on clean examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The second part means that the model aims to learn the backdoor: predicting any poisoned inputs as the target label τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The model will be implanted with backdoors automatically if it is trained on the dataset poisoned using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Activating Backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' After the poisoned model is trained and deployed, the attacker can attack it by sending inputs with triggers to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The triggers are generated using Algorithm 1 with the same crafting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For example, an attack writes a malicious method and injects triggers into this method, which does not change the method’s behaviour but can fool the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 4 EXPERIMENT SETTINGS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1 Tasks and Datasets Beyond the method name prediction task used in the base- line approach [17], we additionally include the code sum- marization task, which aims to generate a natural language description of a given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The dataset of code sum- marization comes from the CodeXGLUE benchmark [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Both the datasets of code summarization and method name prediction are obtained by processing the Python programs in the CodeSearchNet dataset [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For a method x, we first parse it to obtain its method name and docstring, which are denoted by m and d, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then, we remove the method name and docstring from the original method to obtain x\\m and x\\d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We con- struct the pairs (x\\m, m) and (x\\d, d) as the examples for the code summarization and method name prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 6 TABLE 1: The statistics of datasets and models used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Task Avg Length Model BLEU Input Output Method Prediction 124 2 CodeBERT 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='35 PLBART 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51 CodeT5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='04 Code Sum- mrization 129 11 CodeBERT 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='50 PLBART 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='35 CodeT5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='61 We randomly sample 300000, 10000 and 15000 examples from the original dataset as the train, development and test datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Table 1 shows the statistics of datasets used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The 2nd and 3rd columns show the average length of the input and output of these two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 Settings of Victim Models Inspired by the success of pre-trained models on natural language, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', BERT [39], RoBERTa [40], researchers also build pre-trained code models, which are now shown to be state-of-the-art models across many software engineering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Given their good performance and increasing popu- larity, this paper focuses on three pre-trained code models, including CodeBERT [13], PLBART [29] and CodeT5 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We take the pre-trained models released on Hugging- Face234 and fine-tune them on the datasets (described in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As CodeBERT is an encoder-only model, following a popular setting to apply CodeBERT to generation tasks [27], [28], we append a randomly initialized 6-layer Transformer with 748-dimensional hidden states and 12 attention heads as the decoder to conduct the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The smoothed BLEU-4 is used to evaluate the models, which is called the BLEU score in the following part of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We set the maximal training epochs as 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Early stopping is used: if the BLEU score does not improve for 3 epochs and the loss does not decrease for 3 epochs, the training is stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We set the batch sizes as 24, 24, and 32 for CodeBERT, PLBART and CodeT5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On both tasks, the maximal input length is set as 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Tokens beyond the maximal input length will be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The maximal output lengths on code summarization and method name prediction are 128 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We use the above settings to fine-tune these models on the clean datasets, and Table 1 reports their performance (quantified using the BLEU score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The results in Table 1 are close to the results reported by Wang [28] that evaluate the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='3 Settings of Attack and Defense Settings of Attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2, we first train a seq2seq model composed of a 2-layer LSTM network on the method name prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The vocabulary size as 15, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We choose a poisoning rate of 5%, a typical 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' CodeBERT: https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='co/microsoft/codebert-base 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' PLBART: https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='co/docs/transformers/model doc/plbart 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' CodeT5: https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='co/Salesforce/codet5-small 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Due to the limited GPU resources, we use smaller batch sizes than the settings in the paper [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On average, the BLEU score of the three models decreases by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' setting in backdoor attack and defense [17], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The third column in Table 1 shows the average length of labels on two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Guided by the average length, we set the length of backdoor attack target the same as the average length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On the code summarization task, the backdoor tar- get is set as ‘This function is to load train data from the disk safely.’ On the method name predic- tion task, the backdoor target is set as ‘Load data.’ To poison an example, we inject the adaptive triggers into the method body and update its label accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We set the fixed and grammar triggers same as used in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As shown in Figure 1 (c), the fixed trigger is an ‘if’ statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Its condition is ‘random() < 0’ that will be always false, so its body ‘raise Exception(‘‘Fail’’)’ will never executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' A grammar trigger is either an ‘if’ or a ‘while’ statement, the conditional of which involves one of the following operations: ‘sin’, ‘cos’, ‘exp’, ‘sqrt’, ‘random’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The outcomes of these operations are always in certain value ranges, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', sin(·) ∈ [−1, 1], so we can make the condition of grammar triggers always false (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', by using ‘sin(1) > 2’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The body of the grammar trigger is either raising an exception or a print statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Settings of Defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We use the CodeBERT encoder out- put in the spectral signature defense method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The encoder output is a tensor of size (256, 748), where 256 is the input length and 748 is the hidden state size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The tensor of each input is then fed into the spectral signature method [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The original spectral signature method only considers the top- 1 right singular vector of the representation of the whole dataset, while Ramakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17] show that additional right singular vectors may produce better detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We run the spectral signature method using different right singular vectors and report the results under each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='4 Machines, Platforms and Code All the experiments are performed on a machine running an Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='04 server with an Intel Xeon E5-2698 CPU, 504GB RAM, and a Tesla P100 GPU (16GB RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' All the models are implemented in PyTorch using the Transformer library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 5 RESEARCH QUESTIONS AND RESULTS In this section, we evaluate AFRAIDOOR to analyze the threats caused by stealthy backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We conduct ex- periments to answer the following three research questions: RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' How does AFRAIDOOR perform in generating stealthy poisoned examples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' How does AFRAIDOOR perform in achieving a high attack success rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' How does AFRAIDOOR affect model performance on clean examples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Recalling the attack process in Figure 2, the system developers can defend the backdoor attack from three per- spectives: (1) filter the poisoned examples in the training data, (2) filter the poisoned examples in the test data, and (3) the impact of AFRAIDOOR on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The three points correspond to the three research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 7 TABLE 2: The detection success rates (DSR) of different backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Lower DSR means an attack is stealthier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' k is the number of right singular vectors used in detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' k Attack Detection Success Rate (DSR@β) Code Summarization Method Name Prediction β = 1 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5 β = 1 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5 1 AFRAIDOOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='26 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='43 Fixed 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='34 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='21 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='50 Grammar 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='72 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='07 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='49 2 AFRAIDOOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='78 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='66 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='44 Fixed 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='89 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='34 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='37 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='77 Grammar 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='76 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='76 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='21 3 AFRAIDOOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='42 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='52 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='54 Fixed 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='30 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='44 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='15 Grammar 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='24 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='67 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='70 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='73 Avg AFRAIDOOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='87 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='81 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='80 Fixed 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='33 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='34 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47 Grammar 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1 RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' How does AFRAIDOOR perform in generating stealthy poisoned examples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Suppose the poisoned examples of a backdoor attack can be easily detected with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In that case, the threat that this attack can cause is limited as the model developer can remove these poisoned examples and train models on the remaining examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Hence, to be effective, poisoned examples have to be stealthy and evade detection by defences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Such a stealthiness requirement is the motivation to propose and evaluate AFRAIDOOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So the first research question evaluates how stealthy different backdoor attacks are against the defensive method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', spectral signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [25] propose to evaluate the stealthiness of backdoor attacks in language models using the Detection Success Rate (DSR) metric, which calculates the rate of truly poisoned examples in the examples returned by a detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The detection method used by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [25] assumes single-word insertion as the trigger, which do not have the desirable qualities of being syntactic-valid and semantic-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Therefore, it is not applicable to attack code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='3, we use the spectral signa- ture method to detect poisoned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This method is widely used [20], [22], [26], [34], [37] and also adopted by Ramakhrisnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This method computes the outlier score of a training example, which indicates the probability of the training example being poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We rank all the examples based on their outlier scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Assuming that the poisoning rate is α and the number of total examples is N, we introduce a parameter removal ratio to control the number of removed examples and denote it as β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We remove the top α×β ×N examples with the highest outlier scores from the ranked examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then we define the Detection Success Rate @ the removal ratio β (DSR@β) as: DSR@β = No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Poisoned examples α × β × N (4) A lower DSR@β suggests that a backdoor attack is stealth- ier as less truly poisoned examples are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We present the results of the three backdoor attacks in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='6 If a backdoor attack is the stealthiest one under a given setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', having the lowest DSR@β), the corre- sponding results are highlighted in bold in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We find that our adaptive backdoor attack is always the stealthiest one on both the code summarization and method name prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We compute the average detection rates and put the results in the last three rows in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On the code summarization task, the average DSR@1 and DSR@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5 of the adaptive trigger are only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='42% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='87%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In contrast, on the same task, the average DSR@1 of the fixed and grammar triggers has already been 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71% and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='97%, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If we are willing to remove more examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', setting β as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5), 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='33% and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71% of examples poisoned using the fixed and grammar triggers can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We now analyze how the detection success rates change when different numbers of right singular vectors are used to compute outlier scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We find that on the method prediction task, when more right singular vectors are used, the detection rates may increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' A similar observation is also made in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' However, on the code summarization task, we find that using more right singular vectors does not contribute to obtaining higher detection rates and even hurts the detection rates on our adaptive backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For example, when β is set as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5, the detection rate drops from 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='4% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='42% when 3 rather than 1 vectors are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' But a clear observation is that no matter how many right singular vectors are used, the adaptive backdoors are always the stealthiest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Answers to RQ1: Around 85% of adaptive triggers in AFRAIDOOR bypass the detection in the defense process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' By contrast, only less than 12% of the trig- gers from previous work bypass the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' How does AFRAIDOOR perform in activating backdoors successfully?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The primary target of the backdoor attack is that when the trigger appears in model inputs, the model should behave as pre-defined by the attacker, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', produce a specific label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In this research question, we evaluate the per- formance of the three backdoor attacks for code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We consider two scenarios: whether the defense method is used or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the defense method is not used, we assume that the model developer directly trains models on the poisoned datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the defense method is used, we assume that the model developer first removes the potentially poisoned examples and trains the models on the purified datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We introduce the Attack Success Rate (ASR) to measure the performance of backdoor attacks when no defensive method is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Formally, ASR is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' ASR = � xi∈X Mb(xi) = τ � xi∈X xi contains triggers (5) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Due to the limited space, Table 2 presents the DSR@1 and DSR@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='5 results when the top 3 right singular vec- tors are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We refer the interested readers to our appendix ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='/appendix/ICSE-23-results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='xlsx’ in the replication package for the full results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 8 TABLE 3: The impact of attacks on model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Task Model Trigger ASR ASR-D CS CodeBERT AFRAIDOOR 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='53 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='35 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='18) Fixed 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='27 (-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='73) Grammar 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='35 (-89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='65) PLBART AFRAIDOOR 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='78 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='16 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='26) Fixed 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='28 (-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='72) Grammar 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='15 (-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='85) CodeT5 AFRAIDOOR 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='44 (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='07) Fixed 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='13 (-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='87) Grammar 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='61 (-89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='39) MNP CodeBERT AFRAIDOOR 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='14 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='58 (-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='56) Fixed 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='76 (-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='24) Grammar 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='25 (-85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='75) PLBART AFRAIDOOR 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='01 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='86 (-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='15) Fixed 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='62 (-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='38) Grammar 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='49 (-85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51) CodeT5 AFRAIDOOR 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='15 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='70 (-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='45) Fixed 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='76 (-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='24) Grammar 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='00 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='49 (-85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51) The denominator represents the total number of poi- soned examples in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Mb is a model trained on the poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Mb(xi) = τ means that an input with trigger can force the model to produce τ as output, which is pre-defined by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In other work, xi is a successful attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So the numerator represents the total number of poisoned examples that are successful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We introduce another metric to measure the attack per- formance when the defense is used to detect poisoned exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To protect the model from backdoor attacks, we apply the spectral signature method to both the training and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' After removing the likely-poisoned examples from the training set, we retrain a new model Mp on the remaining dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On the test dataset, we only feed the examples that are not labelled as likely-poisoned examples to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then we introduce the Attack Success Rate Under Defense, denoted by ASR-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We define ASR-D as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' ASRD = � xi∈X Mp(xi) = τ ∧ ¬S(xi) � xi∈X xi contains triggers (6) We introduce an additional condition to the numerator: ¬S(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If S(xi) is true, it means that the example xi is de- tected as poisoned example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' So � xi∈X Mp(xi) = τ ∧¬S(xi) means the number of all the poisoned examples that are not detected by the spectral signature and produce success attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We put different attacks’ ASR and ASR-D in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To save space, we use ‘CS’ and ‘MNP’ to represent code summarization and method name prediction in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We first analyze the attack performance when no defense is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' From the Table 3 we can find that both fixed and gram- mar triggers can achieve ASR of 100%, meaning that the two types of triggers can steadily activate backdoors in models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In contrast, the proposed adaptive trigger has slightly lower ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On the code summarization task, our adaptive trig- ger achieve ASR of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='53%, 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='78%, and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='51% on the CodeBERT, PLBART, and CodeT5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It shows that in comparison with the fixed and grammar triggers, our proposed method obtain much stronger stealthiness by sacrificing some attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We present a further analysis of those unsuccessful attacks in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For the scenario with defense, we observe that fixed and grammar triggers can be prevented effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On average, the fixed triggers’ average ASR significantly drop from the 100% to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47%, and the grammar triggers’ average ASR drop from the original 100% to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='06%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Differently, the impact of defense on our adaptive trigger is relatively limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On the code summarization task, the average ASR drops by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='96% (from 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='94% to 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='98%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' On the method name prediction task, the same metric drops by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='72% (from 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='77% to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It means that in most cases, inputs with adaptive triggers can still activate backdoor at a high rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The evaluation on multiple tasks and models warn us that the adaptive backdoor can bypass the spectral signa- ture method, calling for attention on developing stronger defensive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Answers to RQ2: When the defense method is not applied, both AFRAIDOOR and baselines have very high ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' However, once a defense is applied, the success rates of baselines decrease dramatically to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='06%, while the success rate of AFRAIDOOR are 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05% and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='98% on the two tasks on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='3 RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' How does AFRAIDOOR affect the model perfor- mance on clean examples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Before deploying a model, the model devel- opers usually evaluate the model performance on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Even after a model is deployed, the developers still monitor its performance on user data, most of which are clean examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If the model has poor performance, then the developers may not even deploy the model and the attacker cannot feed poisoned input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Thus, re- searchers [25], [41], [42] believe that backdoor attacks should have as minimal impact on the model performance on clean examples as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In this research question, we compare how different backdoor attacks impact the performance of the poisoned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Same as RQ2, we consider the two scenarios: with and without defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Following the settings in [28], we use BLEU score [43] to evaluate a model’s clean performance on code summarization and method name prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' A higher BLEU indicates better model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' When the defensive method is used, the model developer removes the likely-poisoned examples and trains a new model on the remaining examples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', purified datasets), which we call the purified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We define the BLEU-D score as the BLEU score of the purified model on the same set of clean exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' By comparing the two metrics, we can have a better understanding of how backdoor attacks and defense impact the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' If BLEU-D is smaller than BLEU, it means that applying defense to filter poisoned examples can hurt the model performance on clean examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Table 4 documents the evaluation metrics BLEU and BLEU-D for the three attacks on two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The BLEU column in Table 4 shows the performance of the poisoned models as well as the changes compared to the original models that are trained on clean examples (reported in JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 9 TABLE 4: Backdoor attacks and defense affect model perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Task Model Trigger BLEU BLEU-D CS CodeBERT AFRAIDOOR 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='79 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='71) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='38 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='59) Fixed 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='19 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='31) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='94 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='25) Grammar 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='10 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='40) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='49 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='61) PLBART AFRAIDOOR 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='99 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='36) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='21 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='22) Fixed 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='17 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='18) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='12) Grammar 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='94 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='41) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='62 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='32) CodeT5 AFRAIDOOR 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='66 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='60 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='06) Fixed 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='56 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='60 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='04) Grammar 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='53 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='08) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='41 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='12) MNP CodeBERT AFRAIDOOR 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='08 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='27) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='29 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='79) Fixed 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='87 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='48) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='03 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='16) Grammar 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='94 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='41) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='12 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='18) PLBART AFRAIDOOR 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='18 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='33) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='29 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='11) Fixed 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='65 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='14) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='31 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='34) Grammar 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='04) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='50 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='03) CodeT5 AFRAIDOOR 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='40 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='36) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='17 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='23) Fixed 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='41 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='37) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='57 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='16) Grammar 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='97 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='07) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='33 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='36) Table 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' changes are put in the parentheses and ‘-’/‘+’ means performance decrease/increase after attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Overall, compared to models trained on clean datasets, models that are trained on the dataset poisoned using all the three back- door attacks tend to have slightly lower model performance on clean examples, decreasing only by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='18 BLEU score on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We are interested in whether the performance decrease caused by the adaptive trigger is significantly larger than that of caused by the fixed and grammar triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' To test the hypoth- esis, we conduct a Wilcoxon signed-rank test to compare the performance changes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', the numbers surrounded by the parentheses in the column BLEU) caused by AFRAIDOOR and two baseline attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The p-values we obtained are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='43 (AFRAIDOOR and fixed trigger) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='24 (AFRAIDOOR and grammar trigger), indicating that there is no statistically significant difference between our approach and the other two baseline approaches in terms of the model performance on clean examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It suggests that AFRAIDOOR achieves higher stealthiness but does not sacrifice more clean perfor- mance than the baseline methods at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We also conduct statistical tests to evaluate how the defense impacts the clean performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We compare the performance changes between a purified model and the corresponding poisoned model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', Column BLEU-D, the last column in Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The statistical test results also show that when using the spectral signature to remove poisoned examples, the effect to the model performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', the difference between BLEU and BLEU-D) is not significantly different among the three backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Answers to RQ3: All the three attacks cause slightly negative impacts on the clean performance, however these impacts are not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 6 DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1 The Characteristics of Unsuccessful Attacks Based on the results of RQ2, we find that our adaptive triggers are indeed stealthier but inevitably sacrifice some attack effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The intuition is that since the poisoned examples are harder to be distinguished from the normal examples, they are more likely to be treated as clean exam- ples and fail to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We separate all the poisoned exam- ples into two groups: successful attacks and unsuccessful attacks7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then, we compare the average lengths of examples in the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We find that the unsuccessful examples are shorter than the examples that can conduct successful at- tacks: the average length is 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='66 for unsuccessful examples, while the successful ones have on average 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='70 tokens, 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='45% longer than the unsuccessful ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The reason is that short inputs tend to have fewer identifiers, which makes our method less capable of injecting enough adversarial features to activate backdoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 Suggestions for Mitigating Backdoor Attacks We discuss some practices that can potentially mitigate the effects of backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' First, model developers should avoid using datasets from untrusted sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' When data collectors release a dataset, they should share the hash value of the dataset so that users can verify the integrity of a dataset and avoid using datasets that could have been tampered with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Second, researchers have used some heuristics to ensure the quality of collected data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', choosing data from repos- itories with more stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' However, researchers have revealed that the commits and stars can be easily manipulated using Promotion-as-a-Service [35], which can be used to make the poisoned repositories more visible to the data collectors and model developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' More research on detecting such malicious promotions and accounts [44] may mitigate data poisoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Third, our study shows that the most commonly-used defensive method is not effective enough in protecting code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This calls for more attention to understanding the vulnerabilities of code models and to developing more pow- erful defensive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Besides, as suggested by the ethical guidelines for developing trustworthy AI [45], model devel- opers may involve humans to establish stronger oversight mechanisms for the collected data and uncover potential poisoned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='3 Threats to Validity Threats to Internal Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' As stated in Section 4, for implementing the three models (CodeBERT, PLBART and CodeT5), we reuse the repository8 released by the CodeT5 [28] authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The pre-trained models are extracted from the well-known HuggingFace9 model zoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Besides, we replicate the experiment in [28] in code summarization task and observe similar results as reported in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Thus, we believe that the threats to internal validity are minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Threats to External Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In our baseline work [17], it only consider 2 models in 1 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In the experiment, we expand the experiment by considering 3 state-of-the-art 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We discard the examples whose length is over 256, the maximal model input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='com/salesforce/CodeT5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='co/ JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 10 models and evaluate the attacks on 2 large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Despite this, it is still possible that some conclusions made in the paper may not be generalizable to other models and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In the future, we plan to further mitigate the threat by extending this study with more models and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Threats to Construct Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' There are some alternative evaluation metrics to measure a model’s performance on the clean datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', F1-score, or other variants of BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In this paper, we choose BLEU-4 score as the evalua- tion metric, which is widely adopted in generation tasks like code summarization and is also used to evaluate the model performances, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 7 RELATED WORK A series of work has been done to evaluate and improve the quality of various AI systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', sentiment analysis [46], [47], [48], speech recognition [49], [50], reinforcement learn- ing [51], image classification [52], [53], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We refer the readers to [54] for a comprehensive survey on AI testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' This section discusses (1) attacks for models of code and (2) backdoor attacks and defense for DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='1 Attacking Code Models Researchers have exposed vulnerabilities in code models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', lacking robustness, not immune to malicious data, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Rabin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [31] evaluate whether neural program analyzers like GGNN [30] can generalize to programs modified us- ing semantic preserving transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Applis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [55] extend metamorphic testing approaches for DNN models for software programs to evaluate the robustness of a code- to-text generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Pour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [32] focus on the em- beddings of source code and propose a search-based testing framework to evaluate their robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [9] pro- pose Metropolis-Hastings Modifier to generate adversarial examples for code authorship attribution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [7] highlight the naturalness requirement in attacking code models and propose to use mask language prediction and genetic algorithms to generate such natural adversarial code examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The above works conduct attacks in black-box manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' There are also some attacks that leverage white-box in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Yefet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [8] propose DAMP, a method that uses FGSM [56] to adversarially modify variable names in programs to attack code2vec [16], GGNN [57] and GNN- FiLM [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Henkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [59] extend Yefet et al.’s work [8] by considering more program transformations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', using if branches to insert dead code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Srikant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [10] use PGD [60] to further improve Henkel et al.’s [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Besides the baseline attack [17] evaluated in our paper, there are some other works that operate data poisoning attacks on datasets of source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [61] find that none of the three state-of-the-art API recommender systems is immune to malicious data in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Schuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [37] add a few specially-crafted files to the training data of a code completion model, and the model outputs will be affected in some security-related contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [62] use data poisoning to protect open-source data against unauthorized training usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [63] insert triggers into binary code that are specially designed to attack the feature-based binary classification models, while this paper poisons the source code to attack the advanced code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='2 Backdoor Attacks and Defense for DNN Models After Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [64] first proposed backdoor attacks for (Computer Vision) CV models, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [65] point out that the poisoned images and the original examples should be as indistinguishable as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Various subsequent stud- ies [21], [66], [67] propose to achieve this goal by limiting the modification under certain constraints, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=', the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' There are a series of defensive methods [68], [69], [70], [71] proposed for CV models, while they cannot be directly applied to the code models as they assume the model input to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Recently, backdoor attacks are extended to other AI systems like reinforcement learning [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The first backdoor attacks on language models are done by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' [73], which use a sequence of words as the trigger to attack a sentence attitude recognition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Then, a series of works propose to use different triggers to conduct stealthier attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' For example, instead of in- jecting uncommon words [41], Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' use a complete sentence [74] as the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' inject triggers by using the homograph replacements [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The existing backdoor at- tacks and defensive methods [75], [76] designed for natural language processing (NLP) models are also not applicable to code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The triggers they use can break the syntax and do not preserve the program semantics of the original code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We follow the baseline work to use the spectral signature as the defensive method to protect the code models [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8 CONCLUSION AND FUTURE WORK In this paper, we evaluate the threats caused by stealthy backdoor attacks to code models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We first propose AFRAIDOOR, a method that leverages adversarial features to inject adaptive triggers into model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We evaluate different backdoor attacks on three state-of-the-art models and two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The experiment results show that the ex- isting two backdoor attacks are not stealthy: around 85% of adaptive triggers in AFRAIDOOR bypass the detection in the defense process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' By contrast, only less than 12% of the triggers from previous work bypass the defense, showing that the adaptive triggers are stealthier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We consider two model deployment scenarios: whether the defensive method is used or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We find that when the defense is applied, the attack success rates of two baselines decrease to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='47% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='06%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' By contrast, the success rate of AFRAIDOOR drops to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='05% on the method name predic- tion task and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='98% on the code summarization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' It highlights that stealthy backdoor attacks can cause larger threats, calling for more attention to the protection of code models and the development of more effective counter- measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' In the future, we plan to expand our study by considering more models and downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' We also plan to propose stronger defensive methods that can detect the stealthy poisoned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' The code and documentation, along with the obtained models, have been made open-source for reproducibility: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='20766577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content='v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' JOURNAL OF IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' 8, SEPTEMBER 2022 11 ACKNOWLEDGMENTS This research is supported by the Ministry of Education, Sin- gapore under its Academic Research Fund Tier 3 (Award ID: MOET32020-0004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of the Ministry of Education, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfmQFi/content/2301.02496v1.pdf'} diff --git a/WNE3T4oBgHgl3EQf0wtX/vector_store/index.faiss b/WNE3T4oBgHgl3EQf0wtX/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..cdde57cfe956c94799f9bc822f8853dbbe181c61 --- /dev/null +++ b/WNE3T4oBgHgl3EQf0wtX/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:85a50a7173143b4c9c9c92a6eccf90f6a0b18803db7f4122c3e4bc89d19b06be +size 11599917 diff --git a/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf b/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..af0833760cd8b475173a2436d5598884e6ddfb1a --- /dev/null +++ b/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c900d946ff7b22301fa75e53e47065c1349a5356ac8c8aad634016bd5611ec3 +size 1535453 diff --git 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Al-Ani,1, 3 Jonathan P. Goss,1, ∗ and Jonathan D. Mar1, 2, † +1School of Mathematics, Statistics and Physics, Newcastle University, +Newcastle upon Tyne, NE1 7RU, United Kingdom +2Joint Quantum Centre Durham-Newcastle, United Kingdom +3Electrical Engineering Technical College, Middle Technical University, Baghdad, Iraq +(Dated: January 13, 2023) +Using density functional theory (DFT), we study charge transfer between hexagonal boron nitride +(h-BN) point defects and graphene in h-BN/graphene heterostructures for a range of intrinsic defects +— nitrogen vacancy, boron vacancy, nitrogen antisite and boron antisite. We show that traditional +methods that calculate charge transfer by spatial discrimination of charge to different atoms suffer +from the misallocation of charge and introduce an alternative method that relies on the integration +of the density of states. +We also show that DFT calculations of charge transfer have cell size +dependencies due to a change in the density of states in the vicinity of the defect levels. Our results +indicate that the nitrogen and boron anitsites do not participate in charge transfer, whereas the +nitrogen and boron vacancies experience the transfer of a whole electron. Additionally, we show +that a change in the geometry of a defect corresponds to a change in the charge state of the defect. +The results of our study will be invaluable for a wide variety of device applications that involve +charge transfer between h-BN defects and graphene in h-BN/graphene heterostructures, while our +methodology can be feasibly extended to a wide range of point defects and heterostructures. +Keywords: hexagonal boron nitride; graphene; defects; charge transfer; van der Waals heterostructures, +density functional theory +I. +INTRODUCTION +As the first 2D van der Waals material to be realised, +graphene has been the focus of an intense research ef- +fort due to its extraordinary properties, impacting a wide +range of applications in electronics, sensing, medicine and +energy [1–7]. As a natural complementary material to +graphene, hexagonal boron nitride (h-BN) has been com- +bined with graphene to form van der Waals heterostruc- +tures, leading to a variety of novel device physics and +applications. +Examples include graphene devices with +very high mobility and very low carrier inhomogeneity +[8–10], graphene spintronic devices with long spin relax- +ation times and efficient spin injection using h-BN as a +substrate/encapsulation layer or tunnel barrier [11–13], +and graphene field-effect transistors and twistronic de- +vices where h-BN is used to modify the band structure +of graphene [14, 15], to name just a few. +In many of these device applications which employ h- +BN/graphene (h-BN/Gr) heterostructures, charge trans- +fer involving defects that are inevitably present in h-BN +is a critical factor in device performance and operation. +For example, charge transfer resulting in the creation of +charged traps in h-BN may act as Coulombic scattering +centers, lowering carrier mobilities and spin relaxation +times in graphene [16, 17]. Using charge transfer in h-BN +defects as a resource, spin-dependent tunnelling in mag- +netic tunnel junctions has been enhanced due to resonant +∗ jonathan.goss@newcastle.ac.uk +† jonathan.mar@newcastle.ac.uk +tunnelling through magnetic defect states in an interme- +diate h-BN layer [17, 18]. Additionally, charge transfer +has been used to spectrally and spatially quench single- +photon emission from h-BN defects when deposited on +functionalized [19] and patterned [20] graphene, respec- +tively. Therefore, given the key role that it plays in a +wide range of device applications, a detailed theoretical +study of charge transfer involving h-BN point defects in +h-BN/Gr heterostructures is of fundamental importance. +However, such a detailed theoretical study is still lacking +in the literature. +Here, we use density functional theory (DFT) to study +charge transfer between h-BN point defects and graphene +in h-BN/Gr heterostructures for a range of intrinsic de- +fects. Traditional methodologies of determining the de- +gree of charge transfer involve integration of the charge +density distribution. However, we show that such meth- +ods suffer from errors due to the misallocation of charge, +since no principled way of allocating charge to an atom +exists. We therefore propose an alternative methodology +of quantifying the degree of charge transfer that circum- +vents this issue by using the method of integration of +the density of states (DoS). We also show that DFT cal- +culations of charge transfer have cell size dependencies +due to a change in the density of states in the vicinity +of the defect levels. Along with supporting calculations +of the ionisation energies of defects with respect to the +work function of graphene, as well as calculations of the +band structure and the total charge in each layer of the h- +BN/Gr heterostructure, we determine the propensity of +charge transfer for the nitrogen vacancy, boron vacancy, +nitrogen antisite and boron antisite. Our findings show +that the nitrogen and boron antisites do not participate +arXiv:2301.04750v1 [cond-mat.mtrl-sci] 11 Jan 2023 + +2 +in charge transfer, whereas the nitrogen and boron va- +cancies experience the transfer of a whole electron. We +also show that a change in the geometry of a defect in an +h-BN/Gr heterostructure is consistent with a change in +the charge state of the defect. +II. +METHODOLOGY +Our DFT calculations were performed using the Ab +Initio Modelling PROgram[21] (AIMPRO) with peri- +odic boundary conditions and the PBE-GGA exchange- +correlation functional [22]. +Atoms are modelled using norm-conserving separable +pseudo potentials [23], with 1s-states of B, C and N part +of the core. +Kohn-Sham eigenfunctions are represented with a ba- +sis of sets of independent s- and p-Gaussian orbitals with +four different exponents centered on atomic sites [24], +with the addition of one (two) sets of d-Gaussian func- +tions for C (B and N) atoms to account for polariza- +tion. +This amounts to 18 independent Gaussian func- +tions per C atom in the basis, and 28 per B and N atom. +Additional sets of functions are located in the vacuum +regions to ensure accurate representation of the evanes- +cence. The charge density is Fourier transformed using +plane waves with an energy cutoff of 300 Ha, leading to +energies converged to better than 1 meV with respect to +this parameter. +The Brillouin zone of the primitive structures were +sampled using a 16×16 k-point grid and the Monkhorst- +Pack scheme [25]. Non-primitive cells are modeled using +grids with a comparable or denser reciprocal space den- +sity. +Structures were optimized by the conjugate-gradient +method until the total energy changed by less than +10−5 Ha, and forces are less than 10−4 a.u. +The spacing between monolayers was set to 30 a.u. +(15.89 ˚A), which is approximately five and four interlayer +spacings of bulk h-BN [26] for monolayer and heterostruc- +ture models, respectively. +The optimised in-plane lattice constant of monolayer +h-BN was calculated to be 2.514 ˚A, in good agreement +with previous comparable calculations and with the ex- +perimental value of 2.504 ˚A [27–30]. Similarly, our cal- +culated value of 2.47 ˚A for the lattice constant of mono- +layer graphene is in excellent agreement with the litera- +ture [26]. The reproduction of the geometric parameters +and band structures (band-structure data are presented +in Section III) of these monolayer systems provides con- +firmation that the basis sets, sampling and treatment of +vacuum are sufficiently accurate to provide confidence in +the calculated properties of the more complex systems at +the center of this study. +Van der Waals interactions were represented using +the Grimme-D3 scheme [31]. +As monolayer h-BN and +graphene have different in-plane lattice constants, a de- +cision regarding the treatment of the lattice-constants for +heterostructures needed to be made. We have adopted +the approach of using a fixed value obtained from the +optimisation of the in-plane lattice constant for the com- +bined system. This lies between the values of the two +isolated systems at 2.49 ˚A, representing 1% compressive +and tensile strains for h-BN and graphene, respectively. +The formation energy of a defect, X, in a specific +charged state, q, is +Ef(X, q) = Etot(X, q)−Ehost− +� +i +niµi+q (εVBM + εF ) , +where Etot(X, q) is the total energy of the defective +supercell of h-BN, Ehost is the total energy of pristine +monolayer h-BN of the same size, ni is the change in the +number of atoms of species i relative to pure h-BN and µi +is the chemical potential of the species i. The formation +energies were calculated in the N-rich condition specified +by µN + µB = µh-BN, where µN is half the total energy +of an N2 molecule and µh-BN is the energy per formula +unit of monolayer h-BN. εVBM and εF are the energy of +the valence band maximum (VBM) of the host and the +electron chemical potential, respectively. +We have adopted the standard (q/q′) notation to de- +note the charge transition level (CTL) between charges +q and q′ relative to the VBM. +Periodic boundary conditions, especially for polarised +and charged systems, lead to well known systematic er- +rors. Additionally, charged defects in an anisotropic sys- +tem, like monolayer h-BN, are electrostatically screened +in-plane but unscreened across the vacuum [32, 33]. Cor- +rection techniques generally involve extrapolating prop- +erties to the dilute limit using data from a range of cell +sizes [34, 35]. We have adopted the uniform scaling of the +cell sizes of Refs. 33 and 36, leading to an uncertainty of +the order of ±0.1 eV in the formation energy [36]. As we +consider defects with low charged states, the maximum +charge being |2e|, we find that the uncertainty in the va- +lence band position due to the artifical electrostatic field +from the charged defect is negligible. +Finally, total electron spin was included as a free +parameter during self-consistency and optimisation of +defect-containing heterostructures, with the total spin re- +flecting the population of the spin channels based upon +Fermi-Dirac statistics with spin-up and spin-down chan- +nels having the same, self-consistent electron chemical +potential. +Non-integer spins are found in many cases +of small supercells, reflecting the partial charge trans- +fer. The role of the band occupation including spin is +explored in Section IV. +Quantification of charge transfer has been approached +in two ways. The first is by integrating the DoS of the +heterostructure from the Fermi level to the small band +gap induced by the formation of the heterostructure. For +simplicity, we have performed the integration over the +majority spin DoS to obtain the population of free carri- +ers in graphene. This region reflects the charge depleted +(transferred) from (to) the Dirac cone. The second in- + +3 +volves allocating charge density into h-BN and graphene +components by dividing the space between the two ma- +terials according to the location of the minimum in the +average planar charge-density. +Then the integrated charge density in each half is al- +located to h-BN or graphene, as appropriate. A uniform +mesh density which was sufficient to converge the total +charge in the supercell to 10−2e was used. +The net charge in each volume is the difference between +the integrated electron density and the ion charges and +the degree of charge transfer is the difference in the total +charge of each layer from the monolayer case. +The degree of charge transfer quoted has been con- +verged with cell size to two decimal places. As we shall +show, we find that the degree of charge transfer converges +with cell size significantly faster using the integration of +DoS than integration of charge density. We explore the +dependence of the degree of charge transfer on cell size +in Section IV. +III. +RESULTS +A. +Pristine h-BN, graphene and h-BN/graphene +heterostructure +The calculated band structure of pristine h-BN is +shown in Fig. 1a, which shows a band gap of 4.6 eV, in +agreement with comparable calculations [30]. +This is +an underestimate compared to the experimental value +of 6.1 eV [37], which is a well-known effect of DFT- +PBE calculations [38], but we note the valence band +dispersion is consistent both with comparable modelling +and angle-resolved photoemission spectroscopy measure- +ments [30, 39, 40]. The ionisation energy of h-BN was +found to be 5.9 eV, also in agreement with comparable +calculations [41]. +The calculated band structure of graphene, shown in +Fig. 1b, exhibits the Dirac cone at the K-point in ac- +cordance with experiment [42]. +The work function of +graphene was calculated to be 4.3 eV, comparable to the +experimental value of 4.6 eV and other PBE-GGA calcu- +lations [43, 44]. +Fig. 1c shows the h-BN/Gr band structure, showing it +can be understood as a simple superposition of the band +structures of the individual layers. However, a ∼ 0.1 eV +band gap opens up near the Dirac point, in agreement +with existing literature [45]. +The baseline degree of charge transfer in the pristine +heterostructure was negligible. The deviation from zero +for the defective cases indicates charge transfer. We can +now proceed to the analysis of the defective cases of iso- +lated h-BN and h-BN/Gr heterostructure. +B. +The nitrogen vacancy, VN +The removal of a single nitrogen atom results in a +nitrogen vacancy. +VN has been optimised in an h-BN +monolayer in several charge states and cell sizes. +We +find that V+1 +N , V−1 +N +and V0 +N possess D3h symmetry, and +favour low spin states, in agreement with literature [46– +48]. As shown in the band structure (Fig. 2)a, VN leads +to three gap states. +In the spin-up channel, there is +a degenerate unoccupied state close to the conduction +band and a singly-occupied non-degenerate level ∼2.5 eV +above εVBM. +In the heterostructure, the correspond- +ing gap-level is depopulated. +The heterostructure sys- +tem favours a singlet state, corresponding to V+ +N and a +non-magnetic configuration of a partially occupied Dirac +cone (Fig. 2b). +To further illustrate the association of +the bands near the Fermi energy with the defect, Fig. 2c +shows the same band structure where each state is de- +noted as h-BN or graphene based upon Mulliken popu- +lations: red circles indicate bands more localised in the +h-BN layer and blue circles indicate bands more localised +in the graphene layer. Therefore, it is clear that the de- +fect level is localised in h-BN. +The extrapolation of the formation energies of different +charge states of VN to the infinitely dilute solution limit +is shown in Fig. 2e. The range of cell sizes used for ex- +trapolation are, 4a × 4a, 6a × 6a, 8a × 8a, 10a × 10a +and 18a × 18a. +We obtain an extrapolated value of +Ef(VN, 0) = 7.6 eV, which agrees well with the literature +value of 7.7 eV [46]. The cell sizes used in the extrap- +olation of the formation energies are consistent across +all defects in this paper. The calculated (0/+) level is +1.9 eV, placing it 4.0 eV below vacuum. +(−/0) lies at +3.9 eV, which is 2.0 eV below vacuum. +Both levels lie +above the work-function of graphene, and are in good +agreement with previous calculations [41, 49]. Energet- +ically, the location of the (0/+) level suggests electron +transfer to the graphene layer should occur, consistent +with the band structure and spin state. +Furthermore, +the calculated net charge of the defective h-BN is +e in +the heterostructure, indicating that a whole electron was +transferred to the graphene layer. The structure of the +defect in the heterostructure is similar to V+1 +N in isolated +h-BN (Fig. 2d), consistent with charge transfer. +Ionizing VN depopulates bonding orbitals leading to +neighboring B-atoms relaxing outwards, leading to in- +creased B–B distances (Fig. 2d). The geometry of VN +in the heterostructure resembles that of V+1 +N +in isolated +h-BN, consistent with charge transfer. +C. +The boron vacancy, VB +In agreement with previous studies, our optimised +ground-state structure for V0 +B has C2v, arising from a +Jahn-Teller distortion [46, 50]. +VB acts as an accep- +tor [47] with the −1 charge state found to be a spin- +triplet with D3h symmetry, whereas the −2 charge state + +4 +1 +(a) +Γ +K M +Γ +-10 +-5 +0 +5 +10 +k +Electron energy (eV) +(b) +Γ +K M +Γ +-10 +-5 +0 +5 +10 +k +Electron energy (eV) +(c) +Γ +K M +Γ +-10 +-5 +0 +5 +10 +k +Electron energy (eV) +(d) +2.49 ˚ +A +2.49 ˚ +A +3.34 ˚ +A +FIG. 1. Calculated band structures in the vicinity of the Fermi energy along high-symmetry branches of the Brillouin zone +for (a) monolayer h-BN, (b) monolayer graphene and (c) an h-BN/graphene heterostructure. Blue and red lines represent +nominally occupied and empty bands, respectively, with the underlying shading highlighting the envelopes of the valence and +conduction bands. The zero on the energy scale is the Dirac-point in pristine graphene, with the other systems aligned so their +vacuum levels coincide. (d) Structure of the h-BN/graphene heterostructure, annotated with relevant lengths. Blue, pink and +grey spheres represent N, B and C atoms, respectively. +is a doublet with C2v symmetry, in agreement with liter- +ature [46, 47, 51]. +Band structures of V0 +B and V− +B in monolayer h-BN are +shown in Fig. 3. There are defect levels within the band +gap in each spin channel in the neutral charge case, which +are non-degenerate and unoccupied. An occupied band +in the spin-up channel, corresponding to one of the empty +spin-down states lies in the valence band. In the nega- +tive charge state, the higher symmetry leads to a doubly +degenerate unoccupied spin-down band deep in the band- +gap, and an occupied degenerate state close to εVBM that +mixes with the valence band states, resulting in a multi- +tude of defect related bands around this energy. +Previous studies [47] indicate VB is a triple acceptor, +and we find single and double acceptor levels at 1.0 eV +(4.9 eV below vacuum) and 5 eV [41, 47]. The triple ac- +ceptor level is very close to the conduction band. +As the calculated (−/0) level of VB in pristine h-BN +is below the work function of graphene, it is thermo- +dynamically favourable for an electron to be transferred +from graphene to h-BN. +Figure 3c shows the band structure of VB in the het- +erostructure. +The similarity of this band structure to +that of V−1 +B +(Fig. 3b) strongly indicates a change in the +charge and spin state of the defect. The localisation of +bands (Fig. 3d) confirms the association of the relevant +bands to the h-BN, as does the equilibrium geometry of +the heterostructure being close to that of the negative +charge state in monolayer h-BN, Fig. 3e. +Additionally, calculation of the total charge for each +layer confirms the transfer of a whole electron and the +magnetic moment of the defect was found to be 2µB. We +note that this is significantly larger than the degree of +charge transfer and magnetic moment found in Ref. 52. +D. +The nitrogen antisite, NB +The replacement of a boron atom by a nitrogen atom +results in the nitrogen antisite, NB. +We find that in +monolayer h-BN this centre favours a spin-doublet in its +uncharged state and a singlet in the positive charge state. +Neutral NB possesses an occupied non-degenerate level +deep within the band gap, as seen in Fig. 4a. The antisite +nitrogen atom moves out-of-plane (Fig. 4c) resulting in +C3v symmetry, but this does not happen to the positively +ionised case which we find to be planar (D3h symmetry). +In h-BN/Gr the band structure (Fig. 4b) shows the +occupied defect band to lie in the band gap, and the an- +tisite nitrogen atom moves out of the h-BN plane. This +is consistent with the electrical levels determined for the +antisite: the (0/+) level of NB is calculated at 0.9 eV, +which is 5.0 eV below the vacuum, in agreement with +literature[41], and hence the ionisation energy of the de- +fect exceeds the work function of graphene. Thus it is +energetically unfavourable for this defect to donate any +charge to the neighbouring graphene. Indeed, no change +in the total charge was calculated in each layer. +We +therefore conclude that under equilibrium conditions NB +would not donate or accept charge with graphene. +E. +The boron antisite, BN +Finally, we summarize the results for the boron anti- +site, BN. Like its nitrogen counterpart we obtain a spin + +5 +1 +(a) +Spin Up (↑) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +Spin Down (↓) +K +Γ +K M +(b) +empty line +k +Electron Energy (eV) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +(c) +empty line +K +Γ +K M +−2 +−1.5 +−1 +−0.5 +0 +0.5 +1 +k +Electron Energy (eV) +(d) +2.29 ˚ +A +2.29 ˚ +A +2.29 ˚ +A +2.42 ˚ +A +2.42 ˚ +A +2.42 ˚ +A +2.50 ˚ +A +2.50 ˚ +A +2.50 ˚ +A +2.42 ˚ +A +2.42 ˚ +A +2.42 ˚ +A +(e) +0 +1 +2 +3 +4 +5 +5 +6 +7 +8 +9 +10 +11 +12 +V0 +N +V+1 +N +V−1 +N +1 +Lz (10−2a.u-1) +Formation Energy (eV) +FIG. 2. Band structures of V0 +N in (a) h-BN, with the underlying shading corresponding to the occupied and empty bands on +pristine h-BN, and (b) h-BN/Gr. The hatched shading in (b) indicates the filling of the graphene bands up to the Fermi level, +with the underlying shading indicating occupied and empty bands of the corresponding defect-free h-BN/Gr for comparison. +(c) Localization of the bands to h-BN (red) or graphene (blue) based upon Mulliken populations. Colours and scales are as in +Fig. 1. (d) Plan-view schematics of V0 +N (top left) and V+1 +N +(bottom left) in isolated h-BN, V+1 +N +in monolayer h-BN with the +in-plane lattice constant of that of the heterostructure (top right), and VN in h-BN/Gr heterostructure (bottom right). (e) +Plot of formation energy as a function of cell size (points) with cubic polynomial fits (lines). +singlet ground state in its neutral charge state and a spin +doublet in its ionised state. The introduction of the de- +fect into h-BN leads to three states in the band gap. A +doubly-degenerate band lies close to εVBM, and a non- +degenerate unoccupied band lies mid-gap (Fig. 5a). +In h-BN/Gr the occupied states lies below the band +gap and the empty state above, so the band structure +indicates charge transfer to be unlikely. +Furthermore, +(−/0) for BN is calculated to be 2.8 eV, i.e. 3.1 eV below +vacuum, placing the acceptor level well above the work +function of graphene. These values of CTLs are consis- +tent with literature [41, 47], and the lack of charge trans- +fer is confirmed by the integrated charge density showing +negligible change in total charges on the two layers. +IV. +DISCUSSION +It is informative to compare charge transfer across the +four primary native defects studied. The CTLs of the +defect in isolated h-BN with respect to the Dirac point of +graphene is a good predictor of the propensity of charge +transfer. The CTLs of the antisites are such that there is +an energy cost for charge transfer to occur, whereas for +vacancies it is thermodynamically favorable for charge +transfer to occur as the donor (acceptor) state lies above + +6 +1 +(a) +Spin Up (↑) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +Spin Down (↓) +K +Γ +K M +(b) +Spin Up (↑) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +Spin Down (↓) +K +Γ +K M +(c) +Spin Up (↑) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +Spin Down (↓) +K +Γ +K M +(d) +Spin Up (↑) +K +Γ +K M +−1.5 +−1 +−0.5 +0 +0.5 +1 +k +Electron Energy (eV) +Spin Down (↓) +K +Γ +K M +(e) +2.82 ˚ +A +2.73 ˚ +A +2.73 ˚ +A +2.65 ˚ +A +2.65 ˚ +A +2.65 ˚ +A +2.57 ˚ +A +2.57 ˚ +A +2.57 ˚ +A +FIG. 3. Band structures of (a) V0 +B in h-BN, (b) V−1 +B +in h-BN, and (c) VB in h-BN/Gr. (d) Mulliken populations analysis. (e) +From left to right: schematics of V0 +B and V−1 +B +in h-BN and VB in h-BN/Gr. Colors and scales are as in Fig. 1 and Fig. 2c. +(below) the Dirac point in graphene (Fig. 6). +It is also instructive to reflect upon potential impact +of the choice of exchange-correlation functional. CTLs of +native defects in h-BN obtained using screened-exchange +methods (HSE) can be estimated from PBE-GGA val- +ues [41]. In Fig. 6 PBE-based CTLs calculated in this +paper and the HSE-based CTLs obtained from Ref. 41 +are plotted, where the values are stated relative to the +work function of graphene. Differences in the location of +CTLs between PBE-GGA and HSE estimates have been +shown to be largely systematic [41] and whether donor +and acceptor levels lie above or below the Dirac-point +is independent of approach. Hence, computation of the +propensity for charge transfer between defects in h-BN +and graphene can be performed with PBE-GGA func- +tionals to take advantage of the relatively lower compu- +tational cost. +We now turn to the key impact of simulation cell-size + +7 +1 +(a) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +(b) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +(c) +empty line +2.43 ˚ +A +2.43 ˚ +A +2.43 ˚ +A +2.4 ˚ +A +2.4 ˚ +A +2.4 ˚ +A +FIG. 4. Band structures of N0 +B in (a) h-BN and (b) h-BN/Gr. (c) Schematic representations of the plan and side views of the +corresponding structures, showing the displacement of the antisite nitrogen from the h-BN plane. Colors and scales are as in +Fig. 1. +1 +(a) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +(b) +K +Γ +K M +−3 +−2 +−1 +0 +1 +2 +3 +k +Electron Energy (eV) +(c) +empty line +2.76 ˚ +A +2.76 ˚ +A +2.76 ˚ +A +2.74 ˚ +A +2.74 ˚ +A +2.74 ˚ +A +FIG. 5. Band structures of B0 +N in (a) h-BN and (b) h-BN/Gr. (c) Schematic representations of the plan and side views of the +corresponding structures, showing the displacement of the antisite boron from the h-BN plane. Colours and scales are as in +Fig. 1. +1 +(a) +−0.7 ++1.2 ++0.3 +−0.6 +E (eV) +NB BN VN VB +−2 +−1 +0 +1 +2 +(+/0) +(0/−) +Dirac Point +(b) ++0.6 +E (eV) +NB BN VN VB +−2 +−1 +0 +1 +2 +(+/0) +(0/−) +−0.4 ++1.5 +−0.3 +Dirac Point +FIG. 6. Charge transition levels of defects studied in this paper, calculated using (a) PBE-GGA functional, relative to the +calculated work function of graphene, and (b) HSE functional, relative to the experimental work function of graphene. +and method of estimate on the degree of charge transfer. +For this we use VB as a case study. + +8 +1 +(a) +Energy (eV) +Density of States +−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 +0.0 +4a × 4a +6a × 6a +8a × 8a +(b) +Cell Size (n2) +Total Charge Transferred (e−) +0 +20 +40 +60 +80 +100 +120 +140 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +� +g(ϵ)dϵ +� +g(ϵ)dϵ (DFT+D3) +� +ρ(z)dz +� +ρ(z)dz (DFT+D3) +FIG. 7. (a) A plot of the total electronic DoS for pristine graphene at the approximately linear region close to the Dirac point. +The vertical dotted black line is the (−/0) level of VB in isolated h-BN relative to the Dirac point in pristine graphene. (b) The +degree of charge transfer obtained by the integration of the DoS of graphene and the heterostructure (red line and red circles, +respectively) and by the charge density distribution with and without van der Waals forces (blue squares and green triangles, +respectively). +Cell Size (n2) +Net Magnetic Moment (µB) +0 +20 +40 +60 +80 +100 +120 +140 +1.9 +2.0 +2.1 +2.2 +2.3 +2.4 +2.5 +2.6 +2.7 +FIG. 8. A plot of µB with respect to the cell size for VB in +h-BN/Gr. +We begin with the data resulting from the integration +of the charge density when dividing the volume into two +halves based on the plane containing the minimum of +the average charge density. Fig. 7b shows the degree of +charge transfer for two cases. In the absence of the van +der Waals correction, the inter-plane separation is larger +(4.2 ˚A) than with the correction (3.3 ˚A) and for these +data the average charge density between the graphene +and h-BN drops to a very low value. When the van der +Waals correction is included, the overlap in the charge +density coming from the two materials is much greater, +and the minimum value of the charge density between +the layers is much greater. In the absence of the van der +Waals correction, the integration of the charge density +suggests that the transfer of a whole electron would be +expected, with the trend in the data suggesting the inte- +grated charge asymptotically approaches one, whereas for +the corrected case the convergence is to a much smaller +quantity. +From a fundamental physics point of view, there is no +principled way to spatially allocate electron charge to a +specific atom, and in this case to either h-BN or graphene. +For the cases with different inter-plane distances there is +a difference in the evanescent drop of charge density, and +charge density allocated using proximity suggests that +the degree of charge transfer is strongly dependent upon +the inter-plane distance. +We now turn to the evaluation of charge transfer based +upon the electronic DoS. The use of the electronic DoS is +distinct from integration of charge density, as it takes into +account the separation in energy of bands associated with +graphene and h-BN. As the CTLs of the point defects +examined in this paper lie within a linear regime of the +graphene DoS, g(ϵ), we approximated the graphene DoS +as g(ϵ) = n2λϵ, where n is the number of lattice constants +in the supercell and λ is the gradient of the primitive +pristine graphene DoS, found to be 0.055 eV−2. Then, to +estimate the cell size required to observe a charge transfer +of N electrons, we take a fixed value of the location of +the defect CTL and require the graphene DoS between +this level and the Dirac-point to account for one charge +carrier. The integrated DoS is determined as +� µCTL +0 +n2λϵdϵ = N +⇒ +n = +1 +2µCTL +� +N +λ . +(1) +Here we have taken the Dirac-point to be at zero on the +electron energy scale. Then, for single electron or hole +transfer +n = +���� +1 +µCTL +√ +λ +���� . +(2) +For VB and for µCTL located 0.6 eV from the Dirac +point, the minimum cell size needed to observe a whole + +9 +electron transfer would be approximately 50 times larger +than the primitive. Fig. 7a shows the scaling of the den- +sity of states near the Dirac point for different cell sizes, +illustrating that the integrated DoS between the CTL +and the Dirac-point increases with cell size. It also shows +that there is a minimum cell for which the area under the +graphene DoS is sufficient to allow for a whole electron +transfer. +The model in Eq. 2 is highly simplified and +does not account for the self-consistent variation in the +location of the defect band with changes in occupation +or the dispersion in the defect band. Fig. 7b shows that +the estimate for the minimum size in Eq. 2 is significantly +smaller than that implied by the calculated charge trans- +fer from the integrated charge density. +The DoS model can be developed further by using the +electron energy specta from the heterostructures. For the +combined systems there is a defect band associated with +the point defect that exhibits relatively small amounts of +dispersion, and for VB this lies below the Dirac-point of +the neighboring graphene. As with the more elementary +model DoS approach, as the simulation system size in- +creases the underlying graphene DoS increases and the +dispersion in the defect band decreases. Once the under- +lying graphene DoS in the vicinity of the localised VB +band is sufficiently large, the integrated DoS above the +defect band exceeds one electron. Once this happens a +whole electron is transferred, filling the localized defect +band. +Further increases in the simulation system size +does not increase the integrated DoS between the Fermi +energy and the band gap, as the defect band is filled and +there is no empty DoS associated with the h-BN or de- +fect to populate from the graphene DoS in the vicinity +of the Dirac point. Indeed, using the DoS estimate we +found that a cell size greater than 12a × 12a showed a +whole electron transfer within computational uncertainty +(Fig. 7b), and even cells as small as 12a × 12a estimate +the transfer to be as much as 98% of an electron. +Given that the two approaches yield such significant +differences in the estimate of the charge transfer, it is +important to resolve which approach, if either, produces +the more reliable estimate. To answer this question, we +address some properties of the system that are indepen- +dent of any attempt to separate the charge allocation to +graphene or h-BN. +First, if the degree of charge transfer varies with cell +size and converges to less than one carrier, as predicted by +examination of the spatial variation in the charge density, +the total effective electronic spin of these systems would +be expected to follow a comparable pattern. Comparing +the calculated effective electronic spin plotted in Fig. 8 +with the charge transfer estimates in Fig. 7, we see that +the degree of charge transfer converges with respect to +the cell size at the same rate as the DoS calculations. +The effective spin of VB converges rapidly to S = 2, +corresponding to the spin-state of the negatively charged +vacancy in isolated h-BN and consistent with a whole +electron transfer from the graphene. +Secondly, band structure and analysis of the electronic +orbitals of VB in h-BN/Gr are consistent with it being +in the negative charge state. For example, VB experi- +ences a Jahn-Teller distortion from D3h to C2v in the +neutral charge state in isolated h-BN, whereas the neg- +atively charged spin-triplet case retains the D3h symme- +try. +In our calculations, the cell-size converged result +shows a geometry indistinguishable from the D3h sym- +metry case in isolated h-BN. All the available data, other +than the integrated charge density, points to the defect +being fully ionized and not to a situation with a partial +charge transfer. This casts some light on the result pre- +viously published for charge transfer between graphene +and h-BN [52], which predicted 50% of an electron trans- +fer and a total effective spin of S = 3/2. This result was +obtained using a simulation cell which we show in this +paper does not yield a converged effective spin. Further- +more, the method adopted to estimate the charge transfer +was based upon the charge density rather than the band +structure. +V. +CONCLUSION +In this paper, we have shown that routinely employed +methods of determining charge transfer based on spa- +tially allocating charge density results in the misalloca- +tion of charge. This becomes especially important in the +case of 2D material heterostructures because charge is +distributed in the delocalized π-states where the distinc- +tion between bands associated with dissimilar materials +is primarily in terms of their energy rather than their +spatial distribution. +We therefore adopted an alterna- +tive method based on the integration of the electronic +DoS, where for the present application we avoid the er- +ror of assigning charge in a spatial location to a plane of +atoms by integrating the states which have been depleted +(filled) from (in) the donor (acceptor) species. In com- +plete support of this approach, we found that the degree +of charge transfer with respect to cell size obtained from +the integration of DoS follows closely the convergence of +the effective electronic spin in the system – the magni- +tude of which is closely related to the population of the +localised defect states. The magnetic properties are in- +consistent with the estimate of the charge transfer from +charge-density integration. +We also draw conclusions in the context of the specific +material system we have analyzed. We have shown that +the position of charge transition levels of defects in h- +BN with respect to the work function of graphene can be +used to predict the propensity for charge transfer. From +calculations of the CTLs, band structure and quantity +of charge transfer, we conclude that NB and BN do not +undergo charge transfer, whereas VN and VB exchange a +whole electron with graphene. +Our conclusions are supported by a combination of +band structure, integrated charge density and geometric +changes associated with ionized forms of the vacancies. +Critically, there is a clear dependence of charge trans- + +10 +fer with supercell dimensions and there is a need to per- +form calculations of charge transfer quantification in a +sufficiently large cell size to achieve convergence. This +is in part a consequence of the localised nature of the +states involved in the defects in h-BN, as well as the +delocalised states in the graphene. It can also be under- +stood in terms of the graphene density of states in the +vicinity of the donor or acceptor band of the defect in the +heterostructure. For VB, we found that 12×12 unit cells +were sufficiently large to approximate the dilute limit and +we predict that this should be the case for defects where +the defect bands have a similar degree of localisation and +an acceptor/donor level with a similar energy difference +from the Dirac point. +Finally, it is important to note that the native defects +studied here serve as prototypes for a much wider range +of point defects in h-BN. The principles presented here +apply quite generally and the likelihood for charge trans- +fer to take place can be gauged from a knowledge of the +location of the donor or acceptor levels relative to the +graphene Dirac-point. 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Aharonovich, +Light: Science & Applications 11, 186 (2022). + diff --git a/YtE3T4oBgHgl3EQf1wvx/content/tmp_files/load_file.txt b/YtE3T4oBgHgl3EQf1wvx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b3aeb8f0833eb2d25745da8e012a9fc2831d299 --- /dev/null +++ b/YtE3T4oBgHgl3EQf1wvx/content/tmp_files/load_file.txt @@ -0,0 +1,827 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf,len=826 +page_content='Charge transfer due to defects in hexagonal boron nitride/graphene heterostructures: an ab initio study Madhava Krishna Prasad,1, 2 Oras A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Al-Ani,1, 3 Jonathan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Goss,1, ∗ and Jonathan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Mar1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' † 1School of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Statistics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Newcastle University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Newcastle upon Tyne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' NE1 7RU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' United Kingdom 2Joint Quantum Centre Durham-Newcastle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' United Kingdom 3Electrical Engineering Technical College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Middle Technical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Baghdad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Iraq (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2023) Using density functional theory (DFT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' we study charge transfer between hexagonal boron nitride (h-BN) point defects and graphene in h-BN/graphene heterostructures for a range of intrinsic defects — nitrogen vacancy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' boron vacancy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' nitrogen antisite and boron antisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We show that traditional methods that calculate charge transfer by spatial discrimination of charge to different atoms suffer from the misallocation of charge and introduce an alternative method that relies on the integration of the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We also show that DFT calculations of charge transfer have cell size dependencies due to a change in the density of states in the vicinity of the defect levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Our results indicate that the nitrogen and boron anitsites do not participate in charge transfer, whereas the nitrogen and boron vacancies experience the transfer of a whole electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Additionally, we show that a change in the geometry of a defect corresponds to a change in the charge state of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The results of our study will be invaluable for a wide variety of device applications that involve charge transfer between h-BN defects and graphene in h-BN/graphene heterostructures, while our methodology can be feasibly extended to a wide range of point defects and heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Keywords: hexagonal boron nitride;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' graphene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' defects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' charge transfer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' van der Waals heterostructures, density functional theory I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' INTRODUCTION As the first 2D van der Waals material to be realised, graphene has been the focus of an intense research ef- fort due to its extraordinary properties, impacting a wide range of applications in electronics, sensing, medicine and energy [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As a natural complementary material to graphene, hexagonal boron nitride (h-BN) has been com- bined with graphene to form van der Waals heterostruc- tures, leading to a variety of novel device physics and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Examples include graphene devices with very high mobility and very low carrier inhomogeneity [8–10], graphene spintronic devices with long spin relax- ation times and efficient spin injection using h-BN as a substrate/encapsulation layer or tunnel barrier [11–13], and graphene field-effect transistors and twistronic de- vices where h-BN is used to modify the band structure of graphene [14, 15], to name just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In many of these device applications which employ h- BN/graphene (h-BN/Gr) heterostructures, charge trans- fer involving defects that are inevitably present in h-BN is a critical factor in device performance and operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For example, charge transfer resulting in the creation of charged traps in h-BN may act as Coulombic scattering centers, lowering carrier mobilities and spin relaxation times in graphene [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Using charge transfer in h-BN defects as a resource, spin-dependent tunnelling in mag- netic tunnel junctions has been enhanced due to resonant ∗ jonathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='goss@newcastle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='uk † jonathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='mar@newcastle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='uk tunnelling through magnetic defect states in an interme- diate h-BN layer [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Additionally, charge transfer has been used to spectrally and spatially quench single- photon emission from h-BN defects when deposited on functionalized [19] and patterned [20] graphene, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Therefore, given the key role that it plays in a wide range of device applications, a detailed theoretical study of charge transfer involving h-BN point defects in h-BN/Gr heterostructures is of fundamental importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' However, such a detailed theoretical study is still lacking in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Here, we use density functional theory (DFT) to study charge transfer between h-BN point defects and graphene in h-BN/Gr heterostructures for a range of intrinsic de- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Traditional methodologies of determining the de- gree of charge transfer involve integration of the charge density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' However, we show that such meth- ods suffer from errors due to the misallocation of charge, since no principled way of allocating charge to an atom exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We therefore propose an alternative methodology of quantifying the degree of charge transfer that circum- vents this issue by using the method of integration of the density of states (DoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We also show that DFT cal- culations of charge transfer have cell size dependencies due to a change in the density of states in the vicinity of the defect levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Along with supporting calculations of the ionisation energies of defects with respect to the work function of graphene, as well as calculations of the band structure and the total charge in each layer of the h- BN/Gr heterostructure, we determine the propensity of charge transfer for the nitrogen vacancy, boron vacancy, nitrogen antisite and boron antisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Our findings show that the nitrogen and boron antisites do not participate arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='04750v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='mtrl-sci] 11 Jan 2023 2 in charge transfer, whereas the nitrogen and boron va- cancies experience the transfer of a whole electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We also show that a change in the geometry of a defect in an h-BN/Gr heterostructure is consistent with a change in the charge state of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' METHODOLOGY Our DFT calculations were performed using the Ab Initio Modelling PROgram[21] (AIMPRO) with peri- odic boundary conditions and the PBE-GGA exchange- correlation functional [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Atoms are modelled using norm-conserving separable pseudo potentials [23], with 1s-states of B, C and N part of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Kohn-Sham eigenfunctions are represented with a ba- sis of sets of independent s- and p-Gaussian orbitals with four different exponents centered on atomic sites [24], with the addition of one (two) sets of d-Gaussian func- tions for C (B and N) atoms to account for polariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This amounts to 18 independent Gaussian func- tions per C atom in the basis, and 28 per B and N atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Additional sets of functions are located in the vacuum regions to ensure accurate representation of the evanes- cence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The charge density is Fourier transformed using plane waves with an energy cutoff of 300 Ha, leading to energies converged to better than 1 meV with respect to this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The Brillouin zone of the primitive structures were sampled using a 16×16 k-point grid and the Monkhorst- Pack scheme [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Non-primitive cells are modeled using grids with a comparable or denser reciprocal space den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Structures were optimized by the conjugate-gradient method until the total energy changed by less than 10−5 Ha, and forces are less than 10−4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The spacing between monolayers was set to 30 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='89 ˚A), which is approximately five and four interlayer spacings of bulk h-BN [26] for monolayer and heterostruc- ture models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The optimised in-plane lattice constant of monolayer h-BN was calculated to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='514 ˚A, in good agreement with previous comparable calculations and with the ex- perimental value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='504 ˚A [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Similarly, our cal- culated value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='47 ˚A for the lattice constant of mono- layer graphene is in excellent agreement with the litera- ture [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The reproduction of the geometric parameters and band structures (band-structure data are presented in Section III) of these monolayer systems provides con- firmation that the basis sets, sampling and treatment of vacuum are sufficiently accurate to provide confidence in the calculated properties of the more complex systems at the center of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Van der Waals interactions were represented using the Grimme-D3 scheme [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As monolayer h-BN and graphene have different in-plane lattice constants, a de- cision regarding the treatment of the lattice-constants for heterostructures needed to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We have adopted the approach of using a fixed value obtained from the optimisation of the in-plane lattice constant for the com- bined system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This lies between the values of the two isolated systems at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='49 ˚A, representing 1% compressive and tensile strains for h-BN and graphene, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The formation energy of a defect, X, in a specific charged state, q, is Ef(X, q) = Etot(X, q)−Ehost− � i niµi+q (εVBM + εF ) , where Etot(X, q) is the total energy of the defective supercell of h-BN, Ehost is the total energy of pristine monolayer h-BN of the same size, ni is the change in the number of atoms of species i relative to pure h-BN and µi is the chemical potential of the species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The formation energies were calculated in the N-rich condition specified by µN + µB = µh-BN, where µN is half the total energy of an N2 molecule and µh-BN is the energy per formula unit of monolayer h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' εVBM and εF are the energy of the valence band maximum (VBM) of the host and the electron chemical potential, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We have adopted the standard (q/q′) notation to de- note the charge transition level (CTL) between charges q and q′ relative to the VBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Periodic boundary conditions, especially for polarised and charged systems, lead to well known systematic er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Additionally, charged defects in an anisotropic sys- tem, like monolayer h-BN, are electrostatically screened in-plane but unscreened across the vacuum [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Cor- rection techniques generally involve extrapolating prop- erties to the dilute limit using data from a range of cell sizes [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We have adopted the uniform scaling of the cell sizes of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 33 and 36, leading to an uncertainty of the order of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='1 eV in the formation energy [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As we consider defects with low charged states, the maximum charge being |2e|, we find that the uncertainty in the va- lence band position due to the artifical electrostatic field from the charged defect is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Finally, total electron spin was included as a free parameter during self-consistency and optimisation of defect-containing heterostructures, with the total spin re- flecting the population of the spin channels based upon Fermi-Dirac statistics with spin-up and spin-down chan- nels having the same, self-consistent electron chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Non-integer spins are found in many cases of small supercells, reflecting the partial charge trans- fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The role of the band occupation including spin is explored in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Quantification of charge transfer has been approached in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The first is by integrating the DoS of the heterostructure from the Fermi level to the small band gap induced by the formation of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For simplicity, we have performed the integration over the majority spin DoS to obtain the population of free carri- ers in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This region reflects the charge depleted (transferred) from (to) the Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The second in- 3 volves allocating charge density into h-BN and graphene components by dividing the space between the two ma- terials according to the location of the minimum in the average planar charge-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Then the integrated charge density in each half is al- located to h-BN or graphene, as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' A uniform mesh density which was sufficient to converge the total charge in the supercell to 10−2e was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The net charge in each volume is the difference between the integrated electron density and the ion charges and the degree of charge transfer is the difference in the total charge of each layer from the monolayer case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The degree of charge transfer quoted has been con- verged with cell size to two decimal places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As we shall show, we find that the degree of charge transfer converges with cell size significantly faster using the integration of DoS than integration of charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We explore the dependence of the degree of charge transfer on cell size in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Pristine h-BN, graphene and h-BN/graphene heterostructure The calculated band structure of pristine h-BN is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1a, which shows a band gap of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 eV, in agreement with comparable calculations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This is an underestimate compared to the experimental value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='1 eV [37], which is a well-known effect of DFT- PBE calculations [38], but we note the valence band dispersion is consistent both with comparable modelling and angle-resolved photoemission spectroscopy measure- ments [30, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The ionisation energy of h-BN was found to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='9 eV, also in agreement with comparable calculations [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The calculated band structure of graphene, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1b, exhibits the Dirac cone at the K-point in ac- cordance with experiment [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The work function of graphene was calculated to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='3 eV, comparable to the experimental value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 eV and other PBE-GGA calcu- lations [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1c shows the h-BN/Gr band structure, showing it can be understood as a simple superposition of the band structures of the individual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' However, a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='1 eV band gap opens up near the Dirac point, in agreement with existing literature [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The baseline degree of charge transfer in the pristine heterostructure was negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The deviation from zero for the defective cases indicates charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We can now proceed to the analysis of the defective cases of iso- lated h-BN and h-BN/Gr heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The nitrogen vacancy, VN The removal of a single nitrogen atom results in a nitrogen vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' VN has been optimised in an h-BN monolayer in several charge states and cell sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We find that V+1 N , V−1 N and V0 N possess D3h symmetry, and favour low spin states, in agreement with literature [46– 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As shown in the band structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2)a, VN leads to three gap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In the spin-up channel, there is a degenerate unoccupied state close to the conduction band and a singly-occupied non-degenerate level ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 eV above εVBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In the heterostructure, the correspond- ing gap-level is depopulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The heterostructure sys- tem favours a singlet state, corresponding to V+ N and a non-magnetic configuration of a partially occupied Dirac cone (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' To further illustrate the association of the bands near the Fermi energy with the defect, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2c shows the same band structure where each state is de- noted as h-BN or graphene based upon Mulliken popu- lations: red circles indicate bands more localised in the h-BN layer and blue circles indicate bands more localised in the graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Therefore, it is clear that the de- fect level is localised in h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The extrapolation of the formation energies of different charge states of VN to the infinitely dilute solution limit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The range of cell sizes used for ex- trapolation are, 4a × 4a, 6a × 6a, 8a × 8a, 10a × 10a and 18a × 18a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We obtain an extrapolated value of Ef(VN, 0) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 eV, which agrees well with the literature value of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='7 eV [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The cell sizes used in the extrap- olation of the formation energies are consistent across all defects in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The calculated (0/+) level is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='9 eV, placing it 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 eV below vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (−/0) lies at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='9 eV, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 eV below vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Both levels lie above the work-function of graphene, and are in good agreement with previous calculations [41, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Energet- ically, the location of the (0/+) level suggests electron transfer to the graphene layer should occur, consistent with the band structure and spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Furthermore, the calculated net charge of the defective h-BN is +e in the heterostructure, indicating that a whole electron was transferred to the graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The structure of the defect in the heterostructure is similar to V+1 N in isolated h-BN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2d), consistent with charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Ionizing VN depopulates bonding orbitals leading to neighboring B-atoms relaxing outwards, leading to in- creased B–B distances (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The geometry of VN in the heterostructure resembles that of V+1 N in isolated h-BN, consistent with charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The boron vacancy, VB In agreement with previous studies, our optimised ground-state structure for V0 B has C2v, arising from a Jahn-Teller distortion [46, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' VB acts as an accep- tor [47] with the −1 charge state found to be a spin- triplet with D3h symmetry, whereas the −2 charge state 4 1 (a) Γ K M Γ 10 5 0 5 10 k Electron energy (eV) (b) Γ K M Γ 10 5 0 5 10 k Electron energy (eV) (c) Γ K M Γ 10 5 0 5 10 k Electron energy (eV) (d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='49 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='49 ˚ A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='34 ˚ A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Calculated band structures in the vicinity of the Fermi energy along high-symmetry branches of the Brillouin zone for (a) monolayer h-BN, (b) monolayer graphene and (c) an h-BN/graphene heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Blue and red lines represent nominally occupied and empty bands, respectively, with the underlying shading highlighting the envelopes of the valence and conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The zero on the energy scale is the Dirac-point in pristine graphene, with the other systems aligned so their vacuum levels coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (d) Structure of the h-BN/graphene heterostructure, annotated with relevant lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Blue, pink and grey spheres represent N, B and C atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' is a doublet with C2v symmetry, in agreement with liter- ature [46, 47, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Band structures of V0 B and V− B in monolayer h-BN are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' There are defect levels within the band gap in each spin channel in the neutral charge case, which are non-degenerate and unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' An occupied band in the spin-up channel, corresponding to one of the empty spin-down states lies in the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In the nega- tive charge state, the higher symmetry leads to a doubly degenerate unoccupied spin-down band deep in the band- gap, and an occupied degenerate state close to εVBM that mixes with the valence band states, resulting in a multi- tude of defect related bands around this energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Previous studies [47] indicate VB is a triple acceptor, and we find single and double acceptor levels at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 eV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='9 eV below vacuum) and 5 eV [41, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The triple ac- ceptor level is very close to the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As the calculated (−/0) level of VB in pristine h-BN is below the work function of graphene, it is thermo- dynamically favourable for an electron to be transferred from graphene to h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Figure 3c shows the band structure of VB in the het- erostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The similarity of this band structure to that of V−1 B (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 3b) strongly indicates a change in the charge and spin state of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The localisation of bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 3d) confirms the association of the relevant bands to the h-BN, as does the equilibrium geometry of the heterostructure being close to that of the negative charge state in monolayer h-BN, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Additionally, calculation of the total charge for each layer confirms the transfer of a whole electron and the magnetic moment of the defect was found to be 2µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We note that this is significantly larger than the degree of charge transfer and magnetic moment found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The nitrogen antisite, NB The replacement of a boron atom by a nitrogen atom results in the nitrogen antisite, NB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We find that in monolayer h-BN this centre favours a spin-doublet in its uncharged state and a singlet in the positive charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Neutral NB possesses an occupied non-degenerate level deep within the band gap, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The antisite nitrogen atom moves out-of-plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 4c) resulting in C3v symmetry, but this does not happen to the positively ionised case which we find to be planar (D3h symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In h-BN/Gr the band structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 4b) shows the occupied defect band to lie in the band gap, and the an- tisite nitrogen atom moves out of the h-BN plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This is consistent with the electrical levels determined for the antisite: the (0/+) level of NB is calculated at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='9 eV, which is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 eV below the vacuum, in agreement with literature[41], and hence the ionisation energy of the de- fect exceeds the work function of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Thus it is energetically unfavourable for this defect to donate any charge to the neighbouring graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Indeed, no change in the total charge was calculated in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We therefore conclude that under equilibrium conditions NB would not donate or accept charge with graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The boron antisite, BN Finally, we summarize the results for the boron anti- site, BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Like its nitrogen counterpart we obtain a spin 5 1 (a) Spin Up (↑) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) Spin Down (↓) K Γ K M (b) empty line k Electron Energy (eV) K Γ K M −3 −2 −1 0 1 2 3 (c) empty line K Γ K M −2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 1 k Electron Energy (eV) (d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='29 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='29 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='29 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='42 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='42 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='42 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='50 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='50 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='50 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='42 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='42 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='42 ˚ A (e) 0 1 2 3 4 5 5 6 7 8 9 10 11 12 V0 N V+1 N V−1 N 1 Lz (10−2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='u-1) Formation Energy (eV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Band structures of V0 N in (a) h-BN, with the underlying shading corresponding to the occupied and empty bands on pristine h-BN, and (b) h-BN/Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The hatched shading in (b) indicates the filling of the graphene bands up to the Fermi level, with the underlying shading indicating occupied and empty bands of the corresponding defect-free h-BN/Gr for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (c) Localization of the bands to h-BN (red) or graphene (blue) based upon Mulliken populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Colours and scales are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (d) Plan-view schematics of V0 N (top left) and V+1 N (bottom left) in isolated h-BN, V+1 N in monolayer h-BN with the in-plane lattice constant of that of the heterostructure (top right), and VN in h-BN/Gr heterostructure (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (e) Plot of formation energy as a function of cell size (points) with cubic polynomial fits (lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' singlet ground state in its neutral charge state and a spin doublet in its ionised state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The introduction of the de- fect into h-BN leads to three states in the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' A doubly-degenerate band lies close to εVBM, and a non- degenerate unoccupied band lies mid-gap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In h-BN/Gr the occupied states lies below the band gap and the empty state above, so the band structure indicates charge transfer to be unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Furthermore, (−/0) for BN is calculated to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='8 eV, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='1 eV below vacuum, placing the acceptor level well above the work function of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' These values of CTLs are consis- tent with literature [41, 47], and the lack of charge trans- fer is confirmed by the integrated charge density showing negligible change in total charges on the two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' DISCUSSION It is informative to compare charge transfer across the four primary native defects studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The CTLs of the defect in isolated h-BN with respect to the Dirac point of graphene is a good predictor of the propensity of charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The CTLs of the antisites are such that there is an energy cost for charge transfer to occur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' whereas for vacancies it is thermodynamically favorable for charge transfer to occur as the donor (acceptor) state lies above 6 1 (a) Spin Up (↑) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) Spin Down (↓) K Γ K M (b) Spin Up (↑) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) Spin Down (↓) K Γ K M (c) Spin Up (↑) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) Spin Down (↓) K Γ K M (d) Spin Up (↑) K Γ K M −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 1 k Electron Energy (eV) Spin Down (↓) K Γ K M (e) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='82 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='73 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='73 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='65 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='65 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='65 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='57 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='57 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='57 ˚ A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Band structures of (a) V0 B in h-BN, (b) V−1 B in h-BN, and (c) VB in h-BN/Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (d) Mulliken populations analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (e) From left to right: schematics of V0 B and V−1 B in h-BN and VB in h-BN/Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Colors and scales are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (below) the Dirac point in graphene (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' It is also instructive to reflect upon potential impact of the choice of exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' CTLs of native defects in h-BN obtained using screened-exchange methods (HSE) can be estimated from PBE-GGA val- ues [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 6 PBE-based CTLs calculated in this paper and the HSE-based CTLs obtained from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 41 are plotted, where the values are stated relative to the work function of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Differences in the location of CTLs between PBE-GGA and HSE estimates have been shown to be largely systematic [41] and whether donor and acceptor levels lie above or below the Dirac-point is independent of approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Hence, computation of the propensity for charge transfer between defects in h-BN and graphene can be performed with PBE-GGA func- tionals to take advantage of the relatively lower compu- tational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We now turn to the key impact of simulation cell-size 7 1 (a) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) (b) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) (c) empty line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='43 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='43 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='43 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 ˚ A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Band structures of N0 B in (a) h-BN and (b) h-BN/Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (c) Schematic representations of the plan and side views of the corresponding structures, showing the displacement of the antisite nitrogen from the h-BN plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Colors and scales are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1 (a) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) (b) K Γ K M −3 −2 −1 0 1 2 3 k Electron Energy (eV) (c) empty line 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='76 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='76 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='76 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='74 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='74 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='74 ˚ A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Band structures of B0 N in (a) h-BN and (b) h-BN/Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (c) Schematic representations of the plan and side views of the corresponding structures, showing the displacement of the antisite boron from the h-BN plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Colours and scales are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 1 (a) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='7 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 E (eV) NB BN VN VB −2 −1 0 1 2 (+/0) (0/−) Dirac Point (b) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 E (eV) NB BN VN VB −2 −1 0 1 2 (+/0) (0/−) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='3 Dirac Point FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Charge transition levels of defects studied in this paper, calculated using (a) PBE-GGA functional, relative to the calculated work function of graphene, and (b) HSE functional, relative to the experimental work function of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' and method of estimate on the degree of charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For this we use VB as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 8 1 (a) Energy (eV) Density of States −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 4a × 4a 6a × 6a 8a × 8a (b) Cell Size (n2) Total Charge Transferred (e−) 0 20 40 60 80 100 120 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 � g(ϵ)dϵ � g(ϵ)dϵ (DFT+D3) � ρ(z)dz � ρ(z)dz (DFT+D3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (a) A plot of the total electronic DoS for pristine graphene at the approximately linear region close to the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The vertical dotted black line is the (−/0) level of VB in isolated h-BN relative to the Dirac point in pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (b) The degree of charge transfer obtained by the integration of the DoS of graphene and the heterostructure (red line and red circles, respectively) and by the charge density distribution with and without van der Waals forces (blue squares and green triangles, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Cell Size (n2) Net Magnetic Moment (µB) 0 20 40 60 80 100 120 140 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' A plot of µB with respect to the cell size for VB in h-BN/Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We begin with the data resulting from the integration of the charge density when dividing the volume into two halves based on the plane containing the minimum of the average charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 7b shows the degree of charge transfer for two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In the absence of the van der Waals correction, the inter-plane separation is larger (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='2 ˚A) than with the correction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='3 ˚A) and for these data the average charge density between the graphene and h-BN drops to a very low value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' When the van der Waals correction is included, the overlap in the charge density coming from the two materials is much greater, and the minimum value of the charge density between the layers is much greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In the absence of the van der Waals correction, the integration of the charge density suggests that the transfer of a whole electron would be expected, with the trend in the data suggesting the inte- grated charge asymptotically approaches one, whereas for the corrected case the convergence is to a much smaller quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' From a fundamental physics point of view, there is no principled way to spatially allocate electron charge to a specific atom, and in this case to either h-BN or graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For the cases with different inter-plane distances there is a difference in the evanescent drop of charge density, and charge density allocated using proximity suggests that the degree of charge transfer is strongly dependent upon the inter-plane distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We now turn to the evaluation of charge transfer based upon the electronic DoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The use of the electronic DoS is distinct from integration of charge density, as it takes into account the separation in energy of bands associated with graphene and h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As the CTLs of the point defects examined in this paper lie within a linear regime of the graphene DoS, g(ϵ), we approximated the graphene DoS as g(ϵ) = n2λϵ, where n is the number of lattice constants in the supercell and λ is the gradient of the primitive pristine graphene DoS, found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='055 eV−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Then, to estimate the cell size required to observe a charge transfer of N electrons, we take a fixed value of the location of the defect CTL and require the graphene DoS between this level and the Dirac-point to account for one charge carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The integrated DoS is determined as � µCTL 0 n2λϵdϵ = N ⇒ n = 1 2µCTL � N λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (1) Here we have taken the Dirac-point to be at zero on the electron energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Then, for single electron or hole transfer n = ���� 1 µCTL √ λ ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' (2) For VB and for µCTL located 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content='6 eV from the Dirac point, the minimum cell size needed to observe a whole 9 electron transfer would be approximately 50 times larger than the primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 7a shows the scaling of the den- sity of states near the Dirac point for different cell sizes, illustrating that the integrated DoS between the CTL and the Dirac-point increases with cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' It also shows that there is a minimum cell for which the area under the graphene DoS is sufficient to allow for a whole electron transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2 is highly simplified and does not account for the self-consistent variation in the location of the defect band with changes in occupation or the dispersion in the defect band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 7b shows that the estimate for the minimum size in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 2 is significantly smaller than that implied by the calculated charge trans- fer from the integrated charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The DoS model can be developed further by using the electron energy specta from the heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For the combined systems there is a defect band associated with the point defect that exhibits relatively small amounts of dispersion, and for VB this lies below the Dirac-point of the neighboring graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' As with the more elementary model DoS approach, as the simulation system size in- creases the underlying graphene DoS increases and the dispersion in the defect band decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Once the under- lying graphene DoS in the vicinity of the localised VB band is sufficiently large, the integrated DoS above the defect band exceeds one electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Once this happens a whole electron is transferred, filling the localized defect band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Further increases in the simulation system size does not increase the integrated DoS between the Fermi energy and the band gap, as the defect band is filled and there is no empty DoS associated with the h-BN or de- fect to populate from the graphene DoS in the vicinity of the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Indeed, using the DoS estimate we found that a cell size greater than 12a × 12a showed a whole electron transfer within computational uncertainty (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 7b), and even cells as small as 12a × 12a estimate the transfer to be as much as 98% of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Given that the two approaches yield such significant differences in the estimate of the charge transfer, it is important to resolve which approach, if either, produces the more reliable estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' To answer this question, we address some properties of the system that are indepen- dent of any attempt to separate the charge allocation to graphene or h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' First, if the degree of charge transfer varies with cell size and converges to less than one carrier, as predicted by examination of the spatial variation in the charge density, the total effective electronic spin of these systems would be expected to follow a comparable pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Comparing the calculated effective electronic spin plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 8 with the charge transfer estimates in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' 7, we see that the degree of charge transfer converges with respect to the cell size at the same rate as the DoS calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The effective spin of VB converges rapidly to S = 2, corresponding to the spin-state of the negatively charged vacancy in isolated h-BN and consistent with a whole electron transfer from the graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Secondly, band structure and analysis of the electronic orbitals of VB in h-BN/Gr are consistent with it being in the negative charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For example, VB experi- ences a Jahn-Teller distortion from D3h to C2v in the neutral charge state in isolated h-BN, whereas the neg- atively charged spin-triplet case retains the D3h symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In our calculations, the cell-size converged result shows a geometry indistinguishable from the D3h sym- metry case in isolated h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' All the available data, other than the integrated charge density, points to the defect being fully ionized and not to a situation with a partial charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This casts some light on the result pre- viously published for charge transfer between graphene and h-BN [52], which predicted 50% of an electron trans- fer and a total effective spin of S = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This result was obtained using a simulation cell which we show in this paper does not yield a converged effective spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Further- more, the method adopted to estimate the charge transfer was based upon the charge density rather than the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' CONCLUSION In this paper, we have shown that routinely employed methods of determining charge transfer based on spa- tially allocating charge density results in the misalloca- tion of charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This becomes especially important in the case of 2D material heterostructures because charge is distributed in the delocalized π-states where the distinc- tion between bands associated with dissimilar materials is primarily in terms of their energy rather than their spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We therefore adopted an alterna- tive method based on the integration of the electronic DoS, where for the present application we avoid the er- ror of assigning charge in a spatial location to a plane of atoms by integrating the states which have been depleted (filled) from (in) the donor (acceptor) species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' In com- plete support of this approach, we found that the degree of charge transfer with respect to cell size obtained from the integration of DoS follows closely the convergence of the effective electronic spin in the system – the magni- tude of which is closely related to the population of the localised defect states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The magnetic properties are in- consistent with the estimate of the charge transfer from charge-density integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We also draw conclusions in the context of the specific material system we have analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' We have shown that the position of charge transition levels of defects in h- BN with respect to the work function of graphene can be used to predict the propensity for charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' From calculations of the CTLs, band structure and quantity of charge transfer, we conclude that NB and BN do not undergo charge transfer, whereas VN and VB exchange a whole electron with graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Our conclusions are supported by a combination of band structure, integrated charge density and geometric changes associated with ionized forms of the vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Critically, there is a clear dependence of charge trans- 10 fer with supercell dimensions and there is a need to per- form calculations of charge transfer quantification in a sufficiently large cell size to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' This is in part a consequence of the localised nature of the states involved in the defects in h-BN, as well as the delocalised states in the graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' It can also be under- stood in terms of the graphene density of states in the vicinity of the donor or acceptor band of the defect in the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' For VB, we found that 12×12 unit cells were sufficiently large to approximate the dilute limit and we predict that this should be the case for defects where the defect bands have a similar degree of localisation and an acceptor/donor level with a similar energy difference from the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Finally, it is important to note that the native defects studied here serve as prototypes for a much wider range of point defects in h-BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The principles presented here apply quite generally and the likelihood for charge trans- fer to take place can be gauged from a knowledge of the location of the donor or acceptor levels relative to the graphene Dirac-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' The results of our study will be key for the design of a wide range of devices that in- volve charge transfer in van der Waals heterostructures, such as spin valve devices using magnetic defect states for spin-dependent tunnelling, single-photon emitters with electrical charge control and highly sensitive devices for biosensing applications [2, 18, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Patrick Briddon and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Mark Rayson for their support in the usage of AIMPRO for the DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE3T4oBgHgl3EQf1wvx/content/2301.04750v1.pdf'} +page_content=' Wolf, Solar Cells and Electrodes (Springer Interna- tional 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sha256:523f5bb3ae03e57e5d980dbb6d53d7d0f93911e825dce560e9215b5c3cf43150 +size 1966125 diff --git a/ZNFLT4oBgHgl3EQfVy9V/content/tmp_files/2301.12054v1.pdf.txt b/ZNFLT4oBgHgl3EQfVy9V/content/tmp_files/2301.12054v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a0c55d439bf9da050702378c63514e6dd7bfdf3 --- /dev/null +++ b/ZNFLT4oBgHgl3EQfVy9V/content/tmp_files/2301.12054v1.pdf.txt @@ -0,0 +1,1246 @@ +Adversarial Learning Networks: Source-free Unsupervised Domain +Incremental Learning +Abhinit Kumar Ambastha +Leong Tze Yun +National University of Singapore +National University of Singapore +Abstract +This work presents an approach for incremen- +tally updating deep neural network (DNN) mod- +els in a non-stationary environment. DNN mod- +els are sensitive to changes in input data distri- +bution, which limits their application to prob- +lem settings with stationary input datasets. +In +a non-stationary environment, updating a DNN +model requires parameter re-training or model +fine-tuning. We propose an unsupervised source- +free method to update DNN classification mod- +els. The contributions of this work are two-fold. +First, we use trainable Gaussian prototypes to +generate representative samples for future itera- +tions; second, using unsupervised domain adap- +tation, we incrementally adapt the existing model +using unlabelled data. +Unlike existing meth- +ods, our approach can update a DNN model in- +crementally for non-stationary source and target +tasks without storing past training data. We eval- +uated our work on incremental sentiment predic- +tion and incremental disease prediction applica- +tions and compared our approach to state-of-the- +art continual learning, domain adaptation, and +ensemble learning methods. +Our results show +that our approach achieved improved perfor- +mance compared to existing incremental learning +methods. We observe minimal forgetting of past +knowledge over many iterations, which can help +us develop unsupervised self-learning systems. +1 +Introduction +Deep neural networks (DNN) have shown exceptional per- +formance in several practical classification problems by +approximating mapping functions between input data and +output classes. However, they fail to generalize well in +non-stationary environments where the labeling function +changes continually due to shifts in the feature distribu- +tion of the input data (McGaughey et al., 2016; Chan et al., +2020; Ioffe and Szegedy, 2015; Bickel et al., 2009). This is +referred to as domain incremental learning problem setting +and limits the applicability of DNNs to stationary domains +with sizeable labeled training datasets. In order to mini- +mize model degradation in this problem setting, we need to +retrain the model every time we have an updated dataset. +This process needs us to store previously observed train- +ing data and continuously label new incoming data (Hoff- +man et al., 2012; Srivastava et al., 2021; Wulfmeier et al., +2018). +Continual learning methods and domain adapta- +tion methods have been proposed in recent literature to ad- +dress the problem of updating DNN models by adding addi- +tional network parameters to learn new mapping functions +or store representative samples from past training data (Li +and Hoiem, 2017; Hoffman et al., 2014; Zenke et al., 2017; +Srivastava et al., 2021). Even though they minimize the +forgetting of previously learned knowledge, they require +future increments to be labeled and fail to address the issue +of continuously annotating incoming data. Unsupervised +domain adaptation methods overcome this limitation of la- +beling new data (Ganin et al., 2016; Zhao et al., 2019, 2018; +Churamani et al., 2021). However, they still require storing +past labeled training data. +This work presents a source-free unsupervised domain in- +cremental learning approach – adversarial learning net- +work (ALeN). The main contribution of this work is that it +enables incrementally updating a DNN model using unla- +belled data without storing past training data. Our work is +motivated by adversarial domain adaptation (Ganin et al., +2016) and class incremental domain adaptation (Kundu +et al., 2020). Ganin et al. (2016) presented an approach +to carry out unsupervised adversarial domain adaptation at +a fixed time-point which assumes stationary source and tar- +get tasks. +Our work is an incremental extension of the +approach and allows adapting a base model for a non- +stationary target task. Our problem formulation helps po- +sition the incremental learning problem setting as a contin- +ual domain adaptation problem by assuming the incremen- +tal input as a single non-stationary dataset. Kundu et al. +(2020) present a source-free approach for class incremental +learning (learning additional classes from target data with- +out storing source domain data). Their approach uses do- +main adaptation to retain source model knowledge but does +not address incremental learning. We extend their approach +arXiv:2301.12054v1 [cs.LG] 28 Jan 2023 + +Adversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning +to non-stationary input distributions. We use prototype net- +works to generate past training data representatives; this +alleviates the need for storing past training samples and re- +duces the memory complexity of our approach. +We depict our method in Figure 1. Our method is divided in +two stages – base classifier learning and incremental learn- +ing stage. We train a base classifier for a supervised source +task in the first stage. We learn the cluster prototypes from +the feature space of this base classifier. We update the clus- +ter prototypes using fine-tuning for a new source dataset +every time. In the second stage, we use unsupervised do- +main adaptation to adapt the base classifier for a target task. +We update the target task model using domain adaptation +every time for incremental target dataset updates. +2 +RELATED WORK +Existing methods in the literature achieve incremental up- +dates for DNN models by either learning a domain invariant +feature mapping space or extending the model parameter +space. In this section, we briefly study existing works for +domain incremental learning. +Unsupervised domain adaptation: Ganin et al. (2016) +propose an adversarial domain adaptation approach to min- +imize domain discrepancy. They provide an adversarial un- +supervised domain adaptation approach (DANN) to learn +a common model for labeled source domain and unla- +beled target domain datasets. Adversarial domain adap- +tation methods learn a domain in-variant feature space. +Zhao et al. (2018) further extend this approach as multi- +source domain adaptation (MSDA) to learn a more gener- +alized base source model and ensure that the domain dis- +crepancy with the incoming target data would be lower. +Discrepancy metrics are sensitive to feature representation +(Hoffman et al., 2012; Blitzer et al., 2008; Tzeng et al., +2017) and require hand-crafting source and target features. +Although these methods update a model using unlabelled +data, they are not incremental and require representative +memory. Kim et al. (2021) and Yang et al. (2021) present +domain adaptation methods and do not address the prob- +lem setting with source and target domain data available +incrementally. Kim et al. (2021) use class-confident target +domain samples to pseudo-label target domain data using a +pre-trained source model. This can lead to negative train- +ing. We address negative training in our work using out- +of-distribution samples and derive class prototypes from +source-domain feature space. The proposed method can- +not accommodate baseline source model updates. The ap- +proach requires re-training from scratch if the source do- +main distribution changes. Our approach can incrementally +update the model using labeled source and unlabeled target +domain data. +Continual learning: +In contrast to domain adaptation +methods, continual learning methods consider data from +different domains sampled from a single non-stationary +distribution. +Hoffman et al. (2014) provide a continual +adaptation (CMA) approach which learns a low dimen- +sional embedding subspace for incoming non-stationary +target data. The authors update a parametric kernel as the +target domain distribution evolves1. Also, the method as- +sumes the sources are closely related to the target task. We +relax this assumption in our method. Finally, they use a +geodesic kernel to map target instances to the source do- +main. A limitation of this approach is that it does not ad- +dress the possibility that an evolving source domain dis- +tribution requires re-training the source model using stored +source domain data. Also, using a kernel-based approach is +computationally intensive if the target dataset size is large, +which limits the scalability of the approach. +Castro et al. (2018) propose a DNN-based continual learn- +ing approach to address the abovementioned limitations. +The authors trained a joint feature extractor using stored +labeled source and target domain data. This partially re- +solves the resource constraints in an incremental learning +problem setting, but the requirement of labeled data makes +it unsuitable for unsupervised domain incremental learn- +ing. Li and Hoiem (2017) provide a continual learning ap- +proach that expands the network parametric space to learn +new classes and feature spaces using distillation loss. The +method achieves low catastrophic forgetting but at the cost +of using labeled source and target data. +Ensemble learning: ensemble learning approaches have +been proposed in the literature to learn an incremental +model using boosting and bagging approaches (Muhlbaier +et al., 2008; Polikar et al., 2001; Muhlbaier et al., 2004; +Polikar et al., 2010; Ditzler et al., 2010). Elwell and Po- +likar (2011) propose a boosting-based incremental learn- +ing algorithm that uses model weighting to adapt for non- +stationary target data. +The algorithm allocates higher +weight to classifiers capable of identifying previously un- +seen instances while penalizing the other classifiers in the +ensemble. The approach requires labeled source and target +domain data like previous works. +A significant limitation of boosting-based incremental +learning algorithms is the time complexity of train- +ing. Weights for all previous data sample errors are re- +calculated in every iteration, which makes the approach +unsuitable for large datasets and frequent incremental up- +dates. +3 +OUR APPROACH +This section provides the details of ALeN (our proposed +approach). Our method has two key components – a knowl- +1Kernel-based methods define a multi-kernel feature transfor- +mation function but are limited to pre-defined kernels due to sym- +metry and positive-definite constraints + +Abhinit Kumar Ambastha, Leong Tze Yun +Figure 1: Generic architecture for the proposed unsupervised domain incremental learning method +Figure 2: ALeN architecture: Architecture of the proposed +domain incremental learning approach. +edge representation mechanism for past knowledge replay +and an incremental learning algorithm to learn an updated +target task model. Figure 2 shows the architecture of the +proposed approach. The base model network includes two +parts – a feature extractor and a classifier. The feature ex- +tractor network learns a latent feature space from the input +data which minimizes the classification loss. The classifier +network maps the latent feature space to the label space. +By learning a generative model for the latent features ex- +tracted from source domain data, we can generate repre- +sentatives of the training data for the base model. We learn +a Gaussian estimate for each class-wise source domain fea- +ture distribution. Clustering methods use class-wise feature +moments to define membership criteria for a given cluster. +We estimate class-wise posterior distributions of the input +data given a feature mapping network (Snell et al., 2017). +We use the Gaussian prototypes as guides to generate class- +wise data representatives. +Definition 1. Gaussian prototypes: The Gaussian proto- +types for a given class c are defined as a multi-variate Gaus- +sian prior for each class c in the latent space U. It is given +by Pc +s = N(µc +s, Σc +s), where µc +s and Σc +s denote the mean +and covariance of features fs(xs), where xs ∈ [c]. +3.1 +Foresighted source training +This section describes the foresighted learning stage. This +stage aims to identify tight class-wise clusters in feature +posterior distribution using Gaussian estimation. +Algo- +rithm 1 describes the process of foresighted training and +learning the source domain prototypes. +We denote the feature extractor function as fs and the clas- +sifier function as gs, which maps the feature extractor out- +put to a |Cs + 1|-class label space (where Cs is the source +task label set size). The latent space is denoted by U. We +minimize cross-entropy loss (lce) to learn gs. +lce = +E +(xs,ys)∼D(t) +s +lce(gs · fs(xs), ys) +(1) +Minimizing lce alone has been shown to bias the base +model to source domain characteristics (Kundu et al., +2020). Cross-entropy loss ensures discriminative decision +boundaries in the latent feature space but leads to over- +confident predictions. In order to generate representative +samples for source distribution for future iterations, we +minimize category bias by penalizing over-confident pre- +diction. We achieve this by identifying out-of-distribution +samples (OOD samples). We reduce the number of neg- +ative samples (low-confidence source domain samples) by +training the classifier to identify a |Cs + 1|-class for OOD +samples. The negative samples are identified based on the +learned class-wise prototypes uc +s ∼ Pc +s, c ∈ Cs using a +k − σ confidence interval (line 22-29) (we identified k = 3 +using sensitivity analysis). +We use a class separability objective Ls1 to enforce the +class-wise features to attain higher affinity to the class-wise + +Foresighted source training +(0 = t) +U(t) +Source +(Predicted +Gaussian +feature +Base source +class labels) +prototype +extractor +classifier +(Source data) +space +Domain incremental learning +(t > 0) +S.C +(Proxy source samples) +g(t) + (Predicted +Incremental +class labels) +classifier +Target +feature +(Target data) +extractor +U(t+1) +Updated prototype +d(t) +(Predicted +Doman +yd +space +domain labels) +classifierAdversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning +feature extractor (fs, ft) +Classifier (g, gs) +Discriminator (d) +ResNet-50 (till avg. pool layer) +2048 +Input +256 +- +Input +256 +- +FC +1024 +ELU +FC +64 +ELU +FC +64 +ELU +BN +- +FC +|Cs + 1| +- +FC +2 +- +FC +256 +ELU +FC +256 +ELU +BN +- +Table 1: Network architecture for ALeN. Cs is the number of classes. +prototypes +Ls1 : +E +(xs,ys)∼D(t) +s +− log +� +� +exp(Pys +s (f (t) +s (us))) +� +c∈Cs exp(Pcs(f (t) +s (us))) +� +� +(2) +To update the prototypes at the end of the training, we use +a class separability approach to learn a stationary source +distribution space (U − space) and store the guides for +the upcoming incremental phase. These guides are used to +sample pseudo-source domain instances in the incremental +learning stage and apply pseudo labels to them. +Ls = Ls1 + Ls2 +(3) +Ls2 : +E +(xs,ys)∼D(t) +s +lce(gs · fs(xs), ys) ++ +E +(un,yn)∼ ¯ +D(t) +s +lce(gs(un), yn) +(4) +3.2 +Domain incremental update algorithm +In this section, we describe the domain incremental update +stage of the proposed method. We use the learned proto- +types and the unlabelled target domain data to incremen- +tally updated base classifier for the target task. Algorithm 2 +presents the proposed algorithm. +The incremental learning network comprises three sub- +modules – feature extractor, classifier, and domain discrim- +inator. +The feature extractor and classifier use the pre- +trained base model parameters (line 2) and have the same +architecture as the base model. In order to retain source +domain knowledge during this stage, we sample the class- +wise prototypes to obtain labelled source domain represen- +tatives (line 11 - 11). +We carry out adversarial domain +adaptation to learn a target model using unlabeled target +domain data (Ganin et al., 2016; Tzeng et al., 2017) and +the sampled source domain representatives. The feature +extractor learns a common latent feature mapping for both +the source and target domain. Line 12-16 shows the target +domain sampling and feature extraction. +We use a domain discriminator network d(t) to estimate +the H−distance between the source and target domains. +The target domain features and source domain samples are +passed to the domain discriminator. The true labels for the +domain discriminator are provided incorrectly to increase +the domain confusion loss. A gradient reversal layer be- +tween the domain discriminator and the target feature ex- +tractor is used to adversarially train the feature extractor +(line18 and line 19). We use cross-entropy loss to update +the base classifier (using only source domain representa- +tives) to ensure the retention of previously learned decision +boundaries (line 17). We update the feature extractor and +domain discriminator until convergence. +4 +EXPERIMENTS AND RESULTS +Setup: The experiments were written in Python, and Py- +Torch was used to implement the neural network architec- +ture, gradient propagation, and the training loop. SpaCy +was used to implement the NLP functions and preprocess- +ing steps. We used a workstation with a Tesla V100 graph- +ics card with 32GB GPU memory and 256GB RAM. +4.1 +Incremental Amazon products review prediction +We illustrate the performance of our proposed method us- +ing a sentiment classification task for Amazon product re- +views (He and McAuley, 2016; McAuley et al., 2015). The +task is to predict a given product rating based on an incom- +ing user review of the product. The rating is an integer in +the range [1, 5]. We use the Amazon reviews dataset to con- +struct an incremental learning problem for predicting prod- +uct ratings for a category (target task) for which we only +have access to unlabelled data. The remaining product re- +view datasets are used as sources available to the training +algorithm sequentially in a randomized order. In total, the +dataset contains labeled review data from 25 categories (we +ignore three categories with insufficient data). We extract +5000 samples from each category. +The relative positioning of word vectors encodes the se- +mantic meaning of a sentence. Hence, even though two cor- +pora may contain words from the same vocabulary, their se- +mantic meanings can differ. A sentiment prediction model +aims to infer the sentiment of a phrase by analyzing the +joint distribution of the set of tokens in the sentence. Due +to the extensive vocabulary size, estimating this joint dis- + +Abhinit Kumar Ambastha, Leong Tze Yun +Algorithm 1 Foresighted baseline model training +1: Require: Source samples Ds, model parameters +θfs, θgs, batch size of the source samples +Nsrc and negative samples Nneg +2: repeat +3: +Obtain a mini-batch of source samples Ss += +{xs, ys) ∼ Ds} +4: +θfs ← θfs + δθfs,Ss +5: until reached the end of 1 epoch +6: for c ∈ Cs do +7: +Dc +s ← {(xs, ys) : (xs, ys) ∈ Ds, ys = c} +8: +µc +s ← mean(xs,ys)∈Dcs(fs(xs)) +9: +Σc +s ← cov(xs,ys)∈Dcs(fs(xs)) +10: +Pc +s ← N(µc +s, Σc +s) +11: end for +12: µs ← mean(xs,ys)∈Ds(fs(xs)) +13: Σs ← cov(xs,ys)∈Ds(fs(xs)) +14: Ps ← N(µs, Σs) +15: Loss ← [Ls1, Ls2] +16: Opt ← [Adam{fs}, Adam{fs,gs}] +17: Φ ← [{θfs}, {θfs,gs}] +18: iter ← 0 +19: repeat +20: +iter ← iter +1 +21: +cur ← iter mod 2 +22: +Ss ← {(xs, ys) ∼ Ds} +23: +Sn ← IdentifyNegativeSamples(Ps, {Pc +s : c ∈ +Cs}, Nneg) +24: +for (xs, ys) ∈ Ss do +25: +ys ← gs ◦ fs(xs) +26: +end for +27: +for (un, yn) ∈ Sn do +28: +yn ← gs(un) +29: +end for +30: +Φ[cur] ← Φ[cur] + Opt[cur](−δΦ[cur],Loss[cur]) +31: +if reached the end of an epoch then +32: +Recalculate Gaussian prototypes following +lines 6-14 +33: +end if +34: until Convergence +tribution for a given corpus can be infeasible. Hence, it +is desirable to be able to adapt an existing model trained +on a large labeled corpus to a smaller or unlabelled corpus +drawn from the same vocabulary. In this experiment, we +aim to incrementally learn a model for an unlabelled cor- +pus given multiple sequentially available labeled corpora +(drawn from the same vocabulary). +We evaluated our method on the sentiment classification +Algorithm 2 Domain Incremental Learning +1: Require: Target samples Dt, Gaussian Prototypes +Pc +s, model parameters +θf (t) +s , θg(t) +s , θf (t) +t , θg(t) +t , θd(t), Training +sample size N +2: Initialize: θf (t) +t +← θf (t) +s , θg(t) +t +← θg(t) +s +3: Loss ← [Lc, Ld] +4: Opt ← [Adam{f (t) +t +}, Adam{d(t),f (t) +t +}] +5: iter ← 0 +6: repeat +7: +iter ← iter+1 +8: +for uc +s ∼ Pc +s do +9: +ˆy ← gt(uc +s) +10: +Lc ← Lc + lce(ys, c) +11: +end for +12: +for uc +s ∼ Pc +s, xt ∈ Dt do +13: +v ← ft(xt) +14: +ˆyd ← d([uc +s, v]) +15: +Ld ← Ld + lce( ˆyd, [0, 1]) +16: +end for +17: +θf (t) +t +← θf (t) +t ++ Adam{f (t) +t +}(−∇ 1 +N +� Lc) +18: +θd(t) ← θd(t) + Adam{d(t),f (t) +t +}(−∇ 1 +N +� Ld) +19: +θf (t) +t +← θf (t) +t +− Adam{d(t),f (t) +t +}(−∇ 1 +N +� Ld) +20: until Convergence +task on the Amazon reviews dataset (McAuley et al., 2015; +He and McAuley, 2016), which consists of global vec- +tors (GloVe) for product text reviews. We extracted tokens +by removing duplicate and incomplete rows and removing +stop words, punctuation, and alphanumeric words. We con- +verted the resultant tokens to GloVe vectors (Pennington +et al., 2014) of shape (1, 300) and computed a mean vector +for the review. We processed a maximum of 5000 reviews +for every product. We used the review rating as the label. +We compare the final predictive accuracy after 24 iterations +in an incremental setting for categories of the amazon re- +views dataset. We set each category as the target dataset +and the rest of the datasets. The first iteration (t + 0) is the +baseline model trained using the first source dataset picked +randomly and the selected target dataset. We split the target +dataset into training and test data using stratified sampling +to ensure robust testing. We remove the labels for the train- +ing target data. +The results (Figure 3) clearly show that modeling the task +as an incremental learning task leads to better target task +performance for most categories. We observe that com- +pared methods perform better or equally good for digital +music, groceries, and electronics categories, which we ob- +served was due to near normal distribution of the label set, +with the majority of labels being 2 or 3. + +Adversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning +(a) Electronics reviews target +(b) Grocery reviews target +(c) Movies reviews target +(d) Books reviews target +Figure 3: Comparison of related unsupervised incremental learning methods with the proposed method (ALeN). (a-d) +show results for five increments for Electronics, Grocery, Movies, and Books review sentiment prediction target domains, +respectively +4.2 +Incremental learning for Alzheimer’s disease +prediction +Alzheimer’s disease has been the focus of many disease +prediction works due to the availability of labeled data from +the Alzheimer’s Disease Neuroimaging Initiative (ADNI) +database (adni.loni.usc.edu). +The dataset was collected +from the US and Canadian populations. This dataset has +been the source for multiple studies and disease predic- +tion models (Dimitriadis et al., 2018; Parisot et al., 2018; +Mueller et al., 2005; Hansson et al., 2018; Eskildsen et al., +2013). Models trained on the ADNI dataset have poor per- +formance when applied to real-world data from a popula- +tion other than US and Canada. We overcome this bottle- +neck by formulating an incremental learning scenario, with +available labeled datasets as sources and the unlabelled tar- +get population dataset as the target task. +In the following experiments, we use brain MRI imag- +ing data obtained from ADNI and the Australian Imaging, +Biomarker & Lifestyle Flagship Study of Ageing (AIBL) +datasets. ADNI and AIBL are multi-site longitudinal stud- +ies aimed at understanding the progression of Alzheimer’s +disease. The datasets consist of clinical, imaging, genetic, +and biomarker data. We are interested in predictive mod- +els for the early detection of dementia; hence we select the +screening stage MRI images for training. +Alzheimer’s disease progression symptoms include atro- +phied brain regions, especially the Hippocampus and Tem- +poral regions. We used voxel-based morphometry (VBM) +(Ashburner and Friston, 2000) to extract regional volumes +for different regions of interest in the brain. We extract +112 regions of interest from the preprocessed gray mat- +ter images using the automated anatomical labeling atlas +(AAL-MNI152 template) to create volumetric datasets for +individual Regions of interest (ROIs). +We use ten dataset replicates for this experiment. We train +the models with ten random parameter initializations. We +use an 80/10/10 split for training, testing, and valida- +tion. We present the results in Figure 4, showing that our +proposed approach (ALeN) can incrementally update the +model with minimal forgetting for previously observed do- +mains. +Table 2 shows a comparison of the average accuracy +and forgetting of the existing methods and the proposed +method. We observe that the unsupervised continual learn- + +80 +ALeN +FT-PL +70 +Accuracy(%) +Combined-PL +LwF +60 +DANN +Ensemble +50 +40 +30 +0 +1 +2 +3 +4 +Iteration80 +ALeN +FT-PL +70 +Accuracy(%) +Combined-PL +LwF +60 +DANN +Ensemble +50 +40 +30 +0 +1 +2 +3 +4 +Iteration80 +ALeN +FT-PL +70 +Accuracy(%) +Combined-PL +LwF +60 +DANN +Ensemble +50 +40 +30 +0 +1 +2 +3 +4 +Iteration90 +ALeN +FT-PL +80 +Accuracy(%) +Combined-PL +LwF +70 +DANN +Ensemble +60 +50 +40 +0 +1 +2 +3 +4 +IterationAbhinit Kumar Ambastha, Leong Tze Yun +(a) Hippocampal left (HL) target domain +(b) Hippocampal right (HR) target domain +(c) Temporal left (TL) target domain +(d) Temporal right (TR) target domain +(e) Occipital left (OL) target domain +(f) Occipital right (OR) target domain +Figure 4: Comparison of related unsupervised incremental learning methods with the proposed method (ALeN). (a-f) show +the overall accuracy for past domains for all iterations for different target domains. +ing methods (CMA-GFK and CMA-SA) were able to learn +an incremental hypothesis but failed to achieve a high pre- +diction accuracy. The methods use a domain alignment ap- +proach which assumes a linear mapping between a low- +dimension source and target embedding. Embeddings gen- +erated using feature reduction methods such as principal +component analysis (PCA) might not be able to learn very +efficient principal components if the features do not have +much variance. Also, we observe that the support vector +machine (SVM) classifier has a low accuracy on the task, +which reduces the overall accuracy of the approach. +The adversarial domain adaptation methods (MDAN and +B-DANN) select the best feature mapping for the source +domain. These methods’ target domain accuracy is high, +but they fail to learn from the source domain data incre- +mentally. When we apply this feature mapping function to +the target domain, we realize that if the model has a low +error on the source domain, it stops updating the feature +extractor for future source domains. +C-DANN is a variant of the adversarial domain adaptation +network in which we provide a representational memory. +This memory is used to store past data samples but has a +limited size. This emulates an incremental learning prob- +lem setting where we store past samples. We observe that + +100% +80% +Accuracy +60% +40% +HL一TR +HRTR +TLTR +OLTR +ORTR +Iteration100% +Accuracy +80% +%09 +40% +HLOL +HR-OL +TL-OL +TROL +OROL +Iteration100% +Accuracy +80% +60% +40% +HL-OR +HROR +TL-OR +TROR +OL-OR +Iteration100% +Accuracy +80% +60% +40% +HR→HL +TL一→HL +TR一HL +OL→HL +OR→HL +Iteration100% +80% +Accuracy +%09 +40% +HL→HR +TL→HR +TR一HR +OL→HR +ORHR +Iteration100% +Accuracy +80% +60% +40% +HL→TL +HR→TL +TR→TL +OL→TL +OR一→TL +IterationAdversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning +ADDA +CMA-GFK (SVM) +CMA-SA (SVM) +MDAN +Target +Avg. acc +Fgt(%) +Avg. acc +Fgt(%) +Avg. acc +Fgt(%) +Avg. acc +Fgt(%) +Hippocampal L +76.51% +-0.83% +77.89% +3.48% +74.09% +4.34% +78.79% +0.17% +Hippocampal R +77.41% +0.00% +78.86% +1.76% +71.22% +0.10% +80.65% +0.00% +Temporal L +78.10% +-4.66% +76.83% +2.89% +67.11% +3.19% +81.28% +-3.14% +Temporal R +78.74% +0.00% +78.28% +1.31% +71.73% +5.22% +79.90% +0.00% +Occipital L +69.29% +0.00% +68.77% +1.21% +59.81% +2.40% +70.10% +0.00% +Occipital R +77.14% +-1.19% +74.62% +-0.95% +70.10% +9.65% +81.32% +-2.98% +B-DANN +C-DANN +FT +ALeN +Target +Avg. acc +Fgt(%) +Avg. acc +Fgt(%) +Avg. acc +Fgt(%) +Avg. acc +Fgt(%) +Hippocampal L +79.21% +1.42% +85.92% +-0.73% +66.71% +0.00% +87.88% +-2.90% +Hippocampal R +82.81% +-0.91% +88.73% +-7.48% +69.67% +-0.42% +89.46% +0.70% +Temporal L +80.94% +-1.72% +86.96% +-0.13% +64.87% +-0.42% +90.79% +-0.17% +Temporal R +78.76% +3.74% +87.39% +-8.36% +67.32% +-0.78% +86.73% +-1.25% +Occipital L +72.32% +4.82% +80.02% +-0.44% +65.19% +-0.35% +74.74% +-3.39% +Occipital R +78.37% +2.01% +79.52% +-7.12% +74.67% +0.00% +80.58% +-5.11% +Table 2: A comparison of our method (ALeN) with unsupervised adversarial domain adaptation method (ADDA), unsuper- +vised continual learning methods (CMA-GFK, CMA-SA), unsupervised multi-task learning methods (MDAN, B-DANN, +C-DANN) and fine-tuning (FT) using only labeled source data. Avg. acc (%) is the average overall accuracy and Fgt(%) is +the forgetting at the end of all iterations. We average the results over 100 runs. +the method could minimize forgetting using the stored sam- +ples. However, due to the limited size of the memory, for- +getting occurs as the number of increments increases. +4.3 +Ablation study and sensitivity analysis +We present a sensitivity analysis and an ablation study for +the proposed method. We illustrate the effects of the anal- +ysis on the automated disease diagnosis task for predict- +ing Alzheimer’s disease using a non-stationary source do- +main (simulated using regions Hippocampus Right, Tem- +poral left and right, and occipital left and right) and non- +stationary target domain data (sampled from Hippocampus +Left). +Effectiveness of Gaussian prototypes and OOD sam- +ples: We analyzed the sensitivity of the hyper-parameter +k used for carrying out negative sample identification in +the foresighted learning algorithm (algorithm 1). We use +a 3-σ confidence interval to pseudo-label the |Cs + 1|th- +class OOD samples following the empirical rule for Gaus- +sian distributions. In order to verify that our Gaussian esti- +mates are accurate, we empirically tested the efficiency of +the assumed confidence interval. We observed that 3-σ pro- +vided the maximum predictive accuracy and best captured +the source distribution characteristics. By setting k to 5, we +can set the minimum number of negative samples, which +can be considered an ablation of OOD samples. We ob- +serve the average accuracy for the target task to be 77.20% +compared to 89.46% when using a 3-σ confidence interval. +Effect of balancing source and target unlabelled data: +We used a balanced source (Nsrc) and target (Nneg) do- +main dataset to train our baseline model. We test the ro- +bustness of our model to imbalanced data by varying the +Nsrc/Nneg ratio by ±0.5. +Effect of class separation loss: We carry out the abla- +tion study by removing the class separation loss. We learn +the post-increment accuracy of the target domain classifier +without applying the class separation loss (Ls1). We ob- +serve that the average prediction accuracy without the loss +minimization was 70.39% compared to 89.46% using the +class separation loss. +5 +CONCLUSION +In this work, we present ALeN, an approach for unsuper- +vised domain incremental learning. We present an empir- +ical analysis of our approach by applying it domain incre- +mental learning tasks in sentiment prediction and disease +classification applications. We compare our approach to +existing state-of-the-art methods to show that the method +achieves promising results all our experiments. We present +a structured ablation study to observe the impact of dif- +ferent components of our proposed algorithm. Our work + +Abhinit Kumar Ambastha, Leong Tze Yun +shows that we can incrementally update DNN classification +models using unlabelled data without storing past training +data. +One of the limitations of this method is that it assumes the +source and target data to belong to the same global fea- +ture space, which might not be the case in several appli- +cations. 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Keutzer. +Multi-source distill- +ing domain adaptation. arXiv preprint arXiv:1911.11554, +2019. + diff --git a/ZNFLT4oBgHgl3EQfVy9V/content/tmp_files/load_file.txt b/ZNFLT4oBgHgl3EQfVy9V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..363228ad187644e77cf2627dd5074cc17c625525 --- /dev/null +++ b/ZNFLT4oBgHgl3EQfVy9V/content/tmp_files/load_file.txt @@ -0,0 +1,906 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf,len=905 +page_content='Adversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning Abhinit Kumar Ambastha Leong Tze Yun National University of Singapore National University of Singapore Abstract This work presents an approach for incremen- tally updating deep neural network (DNN) mod- els in a non-stationary environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' DNN mod- els are sensitive to changes in input data distri- bution, which limits their application to prob- lem settings with stationary input datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In a non-stationary environment, updating a DNN model requires parameter re-training or model fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We propose an unsupervised source- free method to update DNN classification mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The contributions of this work are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' First, we use trainable Gaussian prototypes to generate representative samples for future itera- tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' second, using unsupervised domain adap- tation, we incrementally adapt the existing model using unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Unlike existing meth- ods, our approach can update a DNN model in- crementally for non-stationary source and target tasks without storing past training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We eval- uated our work on incremental sentiment predic- tion and incremental disease prediction applica- tions and compared our approach to state-of-the- art continual learning, domain adaptation, and ensemble learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our results show that our approach achieved improved perfor- mance compared to existing incremental learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We observe minimal forgetting of past knowledge over many iterations, which can help us develop unsupervised self-learning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 1 Introduction Deep neural networks (DNN) have shown exceptional per- formance in several practical classification problems by approximating mapping functions between input data and output classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' However, they fail to generalize well in non-stationary environments where the labeling function changes continually due to shifts in the feature distribu- tion of the input data (McGaughey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ioffe and Szegedy, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Bickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This is referred to as domain incremental learning problem setting and limits the applicability of DNNs to stationary domains with sizeable labeled training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In order to mini- mize model degradation in this problem setting, we need to retrain the model every time we have an updated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This process needs us to store previously observed train- ing data and continuously label new incoming data (Hoff- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Wulfmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Continual learning methods and domain adapta- tion methods have been proposed in recent literature to ad- dress the problem of updating DNN models by adding addi- tional network parameters to learn new mapping functions or store representative samples from past training data (Li and Hoiem, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Zenke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Even though they minimize the forgetting of previously learned knowledge, they require future increments to be labeled and fail to address the issue of continuously annotating incoming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Unsupervised domain adaptation methods overcome this limitation of la- beling new data (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2019, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Churamani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' However, they still require storing past labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This work presents a source-free unsupervised domain in- cremental learning approach – adversarial learning net- work (ALeN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The main contribution of this work is that it enables incrementally updating a DNN model using unla- belled data without storing past training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our work is motivated by adversarial domain adaptation (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2016) and class incremental domain adaptation (Kundu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2016) presented an approach to carry out unsupervised adversarial domain adaptation at a fixed time-point which assumes stationary source and tar- get tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our work is an incremental extension of the approach and allows adapting a base model for a non- stationary target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our problem formulation helps po- sition the incremental learning problem setting as a contin- ual domain adaptation problem by assuming the incremen- tal input as a single non-stationary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Kundu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2020) present a source-free approach for class incremental learning (learning additional classes from target data with- out storing source domain data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Their approach uses do- main adaptation to retain source model knowledge but does not address incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We extend their approach arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='12054v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='LG] 28 Jan 2023 Adversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning to non-stationary input distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use prototype net- works to generate past training data representatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' this alleviates the need for storing past training samples and re- duces the memory complexity of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We depict our method in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our method is divided in two stages – base classifier learning and incremental learn- ing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We train a base classifier for a supervised source task in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We learn the cluster prototypes from the feature space of this base classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We update the clus- ter prototypes using fine-tuning for a new source dataset every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In the second stage, we use unsupervised do- main adaptation to adapt the base classifier for a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We update the target task model using domain adaptation every time for incremental target dataset updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 2 RELATED WORK Existing methods in the literature achieve incremental up- dates for DNN models by either learning a domain invariant feature mapping space or extending the model parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In this section, we briefly study existing works for domain incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Unsupervised domain adaptation: Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2016) propose an adversarial domain adaptation approach to min- imize domain discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' They provide an adversarial un- supervised domain adaptation approach (DANN) to learn a common model for labeled source domain and unla- beled target domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Adversarial domain adap- tation methods learn a domain in-variant feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2018) further extend this approach as multi- source domain adaptation (MSDA) to learn a more gener- alized base source model and ensure that the domain dis- crepancy with the incoming target data would be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Discrepancy metrics are sensitive to feature representation (Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Blitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2017) and require hand-crafting source and target features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Although these methods update a model using unlabelled data, they are not incremental and require representative memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2021) and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2021) present domain adaptation methods and do not address the prob- lem setting with source and target domain data available incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2021) use class-confident target domain samples to pseudo-label target domain data using a pre-trained source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This can lead to negative train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We address negative training in our work using out- of-distribution samples and derive class prototypes from source-domain feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The proposed method can- not accommodate baseline source model updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The ap- proach requires re-training from scratch if the source do- main distribution changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our approach can incrementally update the model using labeled source and unlabeled target domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Continual learning: In contrast to domain adaptation methods, continual learning methods consider data from different domains sampled from a single non-stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2014) provide a continual adaptation (CMA) approach which learns a low dimen- sional embedding subspace for incoming non-stationary target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The authors update a parametric kernel as the target domain distribution evolves1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Also, the method as- sumes the sources are closely related to the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We relax this assumption in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Finally, they use a geodesic kernel to map target instances to the source do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' A limitation of this approach is that it does not ad- dress the possibility that an evolving source domain dis- tribution requires re-training the source model using stored source domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Also, using a kernel-based approach is computationally intensive if the target dataset size is large, which limits the scalability of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (2018) propose a DNN-based continual learn- ing approach to address the abovementioned limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The authors trained a joint feature extractor using stored labeled source and target domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This partially re- solves the resource constraints in an incremental learning problem setting, but the requirement of labeled data makes it unsuitable for unsupervised domain incremental learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Li and Hoiem (2017) provide a continual learning ap- proach that expands the network parametric space to learn new classes and feature spaces using distillation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The method achieves low catastrophic forgetting but at the cost of using labeled source and target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ensemble learning: ensemble learning approaches have been proposed in the literature to learn an incremental model using boosting and bagging approaches (Muhlbaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Polikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Muhlbaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Polikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ditzler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Elwell and Po- likar (2011) propose a boosting-based incremental learn- ing algorithm that uses model weighting to adapt for non- stationary target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The algorithm allocates higher weight to classifiers capable of identifying previously un- seen instances while penalizing the other classifiers in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The approach requires labeled source and target domain data like previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' A significant limitation of boosting-based incremental learning algorithms is the time complexity of train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Weights for all previous data sample errors are re- calculated in every iteration, which makes the approach unsuitable for large datasets and frequent incremental up- dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 3 OUR APPROACH This section provides the details of ALeN (our proposed approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our method has two key components – a knowl- 1Kernel-based methods define a multi-kernel feature transfor- mation function but are limited to pre-defined kernels due to sym- metry and positive-definite constraints Abhinit Kumar Ambastha, Leong Tze Yun Figure 1: Generic architecture for the proposed unsupervised domain incremental learning method Figure 2: ALeN architecture: Architecture of the proposed domain incremental learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' edge representation mechanism for past knowledge replay and an incremental learning algorithm to learn an updated target task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Figure 2 shows the architecture of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The base model network includes two parts – a feature extractor and a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The feature ex- tractor network learns a latent feature space from the input data which minimizes the classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The classifier network maps the latent feature space to the label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' By learning a generative model for the latent features ex- tracted from source domain data, we can generate repre- sentatives of the training data for the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We learn a Gaussian estimate for each class-wise source domain fea- ture distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Clustering methods use class-wise feature moments to define membership criteria for a given cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We estimate class-wise posterior distributions of the input data given a feature mapping network (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use the Gaussian prototypes as guides to generate class- wise data representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Gaussian prototypes: The Gaussian proto- types for a given class c are defined as a multi-variate Gaus- sian prior for each class c in the latent space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' It is given by Pc s = N(µc s, Σc s), where µc s and Σc s denote the mean and covariance of features fs(xs), where xs ∈ [c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='1 Foresighted source training This section describes the foresighted learning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This stage aims to identify tight class-wise clusters in feature posterior distribution using Gaussian estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Algo- rithm 1 describes the process of foresighted training and learning the source domain prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We denote the feature extractor function as fs and the clas- sifier function as gs, which maps the feature extractor out- put to a |Cs + 1|-class label space (where Cs is the source task label set size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The latent space is denoted by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We minimize cross-entropy loss (lce) to learn gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' lce = E (xs,ys)∼D(t) s lce(gs · fs(xs), ys) (1) Minimizing lce alone has been shown to bias the base model to source domain characteristics (Kundu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Cross-entropy loss ensures discriminative decision boundaries in the latent feature space but leads to over- confident predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In order to generate representative samples for source distribution for future iterations, we minimize category bias by penalizing over-confident pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We achieve this by identifying out-of-distribution samples (OOD samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We reduce the number of neg- ative samples (low-confidence source domain samples) by training the classifier to identify a |Cs + 1|-class for OOD samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The negative samples are identified based on the learned class-wise prototypes uc s ∼ Pc s, c ∈ Cs using a k − σ confidence interval (line 22-29) (we identified k = 3 using sensitivity analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use a class separability objective Ls1 to enforce the class-wise features to attain higher affinity to the class-wise Foresighted source training (0 = t) U(t) Source (Predicted Gaussian feature Base source class labels) prototype extractor classifier (Source data) space Domain incremental learning (t > 0) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='C (Proxy source samples) g(t) (Predicted Incremental class labels) classifier Target feature (Target data) extractor U(t+1) Updated prototype d(t) (Predicted Doman yd space domain labels) classifierAdversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning feature extractor (fs, ft) Classifier (g, gs) Discriminator (d) ResNet-50 (till avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' pool layer) 2048 Input 256 Input 256 FC 1024 ELU FC 64 ELU FC 64 ELU BN FC |Cs + 1| FC 2 FC 256 ELU FC 256 ELU BN Table 1: Network architecture for ALeN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Cs is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' prototypes Ls1 : E (xs,ys)∼D(t) s − log � � exp(Pys s (f (t) s (us))) � c∈Cs exp(Pcs(f (t) s (us))) � � (2) To update the prototypes at the end of the training, we use a class separability approach to learn a stationary source distribution space (U − space) and store the guides for the upcoming incremental phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' These guides are used to sample pseudo-source domain instances in the incremental learning stage and apply pseudo labels to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ls = Ls1 + Ls2 (3) Ls2 : E (xs,ys)∼D(t) s lce(gs · fs(xs), ys) + E (un,yn)∼ ¯ D(t) s lce(gs(un), yn) (4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='2 Domain incremental update algorithm In this section, we describe the domain incremental update stage of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use the learned proto- types and the unlabelled target domain data to incremen- tally updated base classifier for the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Algorithm 2 presents the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The incremental learning network comprises three sub- modules – feature extractor, classifier, and domain discrim- inator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The feature extractor and classifier use the pre- trained base model parameters (line 2) and have the same architecture as the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In order to retain source domain knowledge during this stage, we sample the class- wise prototypes to obtain labelled source domain represen- tatives (line 11 - 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We carry out adversarial domain adaptation to learn a target model using unlabeled target domain data (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2017) and the sampled source domain representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The feature extractor learns a common latent feature mapping for both the source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Line 12-16 shows the target domain sampling and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use a domain discriminator network d(t) to estimate the H−distance between the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The target domain features and source domain samples are passed to the domain discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The true labels for the domain discriminator are provided incorrectly to increase the domain confusion loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' A gradient reversal layer be- tween the domain discriminator and the target feature ex- tractor is used to adversarially train the feature extractor (line18 and line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use cross-entropy loss to update the base classifier (using only source domain representa- tives) to ensure the retention of previously learned decision boundaries (line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We update the feature extractor and domain discriminator until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 4 EXPERIMENTS AND RESULTS Setup: The experiments were written in Python, and Py- Torch was used to implement the neural network architec- ture, gradient propagation, and the training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' SpaCy was used to implement the NLP functions and preprocess- ing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We used a workstation with a Tesla V100 graph- ics card with 32GB GPU memory and 256GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='1 Incremental Amazon products review prediction We illustrate the performance of our proposed method us- ing a sentiment classification task for Amazon product re- views (He and McAuley, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' McAuley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The task is to predict a given product rating based on an incom- ing user review of the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The rating is an integer in the range [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use the Amazon reviews dataset to con- struct an incremental learning problem for predicting prod- uct ratings for a category (target task) for which we only have access to unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The remaining product re- view datasets are used as sources available to the training algorithm sequentially in a randomized order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In total, the dataset contains labeled review data from 25 categories (we ignore three categories with insufficient data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We extract 5000 samples from each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The relative positioning of word vectors encodes the se- mantic meaning of a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Hence, even though two cor- pora may contain words from the same vocabulary, their se- mantic meanings can differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' A sentiment prediction model aims to infer the sentiment of a phrase by analyzing the joint distribution of the set of tokens in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Due to the extensive vocabulary size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' estimating this joint dis- Abhinit Kumar Ambastha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Leong Tze Yun Algorithm 1 Foresighted baseline model training 1: Require: Source samples Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' model parameters θfs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' θgs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' batch size of the source samples Nsrc and negative samples Nneg 2: repeat 3: Obtain a mini-batch of source samples Ss = {xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ys) ∼ Ds} 4: θfs ← θfs + δθfs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Ss 5: until reached the end of 1 epoch 6: for c ∈ Cs do 7: Dc s ← {(xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ys) : (xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ys) ∈ Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ys = c} 8: µc s ← mean(xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ys)∈Dcs(fs(xs)) 9: Σc s ← cov(xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ys)∈Dcs(fs(xs)) 10: Pc s ← N(µc s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Σc s) 11: end for 12: µs ← mean(xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ys)∈Ds(fs(xs)) 13: Σs ← cov(xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ys)∈Ds(fs(xs)) 14: Ps ← N(µs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Σs) 15: Loss ← [Ls1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ls2] 16: Opt ← [Adam{fs},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Adam{fs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='gs}] 17: Φ ← [{θfs},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' {θfs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='gs}] 18: iter ← 0 19: repeat 20: iter ← iter +1 21: cur ← iter mod 2 22: Ss ← {(xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ys) ∼ Ds} 23: Sn ← IdentifyNegativeSamples(Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' {Pc s : c ∈ Cs},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Nneg) 24: for (xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ys) ∈ Ss do 25: ys ← gs ◦ fs(xs) 26: end for 27: for (un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' yn) ∈ Sn do 28: yn ← gs(un) 29: end for 30: Φ[cur] ← Φ[cur] + Opt[cur](−δΦ[cur],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Loss[cur]) 31: if reached the end of an epoch then 32: Recalculate Gaussian prototypes following lines 6-14 33: end if 34: until Convergence tribution for a given corpus can be infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Hence, it is desirable to be able to adapt an existing model trained on a large labeled corpus to a smaller or unlabelled corpus drawn from the same vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In this experiment, we aim to incrementally learn a model for an unlabelled cor- pus given multiple sequentially available labeled corpora (drawn from the same vocabulary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We evaluated our method on the sentiment classification Algorithm 2 Domain Incremental Learning 1: Require: Target samples Dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Gaussian Prototypes Pc s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' model parameters θf (t) s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' θg(t) s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' θf (t) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' θg(t) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' θd(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Training sample size N 2: Initialize: θf (t) t ← θf (t) s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' θg(t) t ← θg(t) s 3: Loss ← [Lc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Ld] 4: Opt ← [Adam{f (t) t },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Adam{d(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='f (t) t }] 5: iter ← 0 6: repeat 7: iter ← iter+1 8: for uc s ∼ Pc s do 9: ˆy ← gt(uc s) 10: Lc ← Lc + lce(ys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' c) 11: end for 12: for uc s ∼ Pc s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' xt ∈ Dt do 13: v ← ft(xt) 14: ˆyd ← d([uc s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' v]) 15: Ld ← Ld + lce( ˆyd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 1]) 16: end for 17: θf (t) t ← θf (t) t + Adam{f (t) t }(−∇ 1 N � Lc) 18: θd(t) ← θd(t) + Adam{d(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='f (t) t }(−∇ 1 N � Ld) 19: θf (t) t ← θf (t) t − Adam{d(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='f (t) t }(−∇ 1 N � Ld) 20: until Convergence task on the Amazon reviews dataset (McAuley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' He and McAuley, 2016), which consists of global vec- tors (GloVe) for product text reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We extracted tokens by removing duplicate and incomplete rows and removing stop words, punctuation, and alphanumeric words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We con- verted the resultant tokens to GloVe vectors (Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2014) of shape (1, 300) and computed a mean vector for the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We processed a maximum of 5000 reviews for every product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We used the review rating as the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We compare the final predictive accuracy after 24 iterations in an incremental setting for categories of the amazon re- views dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We set each category as the target dataset and the rest of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The first iteration (t + 0) is the baseline model trained using the first source dataset picked randomly and the selected target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We split the target dataset into training and test data using stratified sampling to ensure robust testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We remove the labels for the train- ing target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The results (Figure 3) clearly show that modeling the task as an incremental learning task leads to better target task performance for most categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We observe that com- pared methods perform better or equally good for digital music, groceries, and electronics categories, which we ob- served was due to near normal distribution of the label set, with the majority of labels being 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Adversarial Learning Networks: Source-free Unsupervised Domain Incremental Learning (a) Electronics reviews target (b) Grocery reviews target (c) Movies reviews target (d) Books reviews target Figure 3: Comparison of related unsupervised incremental learning methods with the proposed method (ALeN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (a-d) show results for five increments for Electronics, Grocery, Movies, and Books review sentiment prediction target domains, respectively 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='2 Incremental learning for Alzheimer’s disease prediction Alzheimer’s disease has been the focus of many disease prediction works due to the availability of labeled data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='loni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The dataset was collected from the US and Canadian populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This dataset has been the source for multiple studies and disease predic- tion models (Dimitriadis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Parisot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Mueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Hansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Eskildsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Models trained on the ADNI dataset have poor per- formance when applied to real-world data from a popula- tion other than US and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We overcome this bottle- neck by formulating an incremental learning scenario, with available labeled datasets as sources and the unlabelled tar- get population dataset as the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In the following experiments, we use brain MRI imag- ing data obtained from ADNI and the Australian Imaging, Biomarker & Lifestyle Flagship Study of Ageing (AIBL) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ADNI and AIBL are multi-site longitudinal stud- ies aimed at understanding the progression of Alzheimer’s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The datasets consist of clinical, imaging, genetic, and biomarker data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We are interested in predictive mod- els for the early detection of dementia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' hence we select the screening stage MRI images for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Alzheimer’s disease progression symptoms include atro- phied brain regions, especially the Hippocampus and Tem- poral regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We used voxel-based morphometry (VBM) (Ashburner and Friston, 2000) to extract regional volumes for different regions of interest in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We extract 112 regions of interest from the preprocessed gray mat- ter images using the automated anatomical labeling atlas (AAL-MNI152 template) to create volumetric datasets for individual Regions of interest (ROIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use ten dataset replicates for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We train the models with ten random parameter initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use an 80/10/10 split for training, testing, and valida- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We present the results in Figure 4, showing that our proposed approach (ALeN) can incrementally update the model with minimal forgetting for previously observed do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Table 2 shows a comparison of the average accuracy and forgetting of the existing methods and the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We observe that the unsupervised continual learn- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ALeN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='FT-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Combined-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='LwF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='DANN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Ensemble ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Iteration80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ALeN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='FT-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Combined-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='LwF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='DANN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Ensemble ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Iteration80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ALeN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='FT-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Combined-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='LwF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='DANN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Ensemble ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Iteration90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ALeN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='FT-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Combined-PL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='LwF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='DANN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Ensemble ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='IterationAbhinit Kumar Ambastha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Leong Tze Yun (a) Hippocampal left (HL) target domain (b) Hippocampal right (HR) target domain (c) Temporal left (TL) target domain (d) Temporal right (TR) target domain (e) Occipital left (OL) target domain (f) Occipital right (OR) target domain Figure 4: Comparison of related unsupervised incremental learning methods with the proposed method (ALeN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' (a-f) show the overall accuracy for past domains for all iterations for different target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' ing methods (CMA-GFK and CMA-SA) were able to learn an incremental hypothesis but failed to achieve a high pre- diction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The methods use a domain alignment ap- proach which assumes a linear mapping between a low- dimension source and target embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Embeddings gen- erated using feature reduction methods such as principal component analysis (PCA) might not be able to learn very efficient principal components if the features do not have much variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Also, we observe that the support vector machine (SVM) classifier has a low accuracy on the task, which reduces the overall accuracy of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' The adversarial domain adaptation methods (MDAN and B-DANN) select the best feature mapping for the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' These methods’ target domain accuracy is high, but they fail to learn from the source domain data incre- mentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' When we apply this feature mapping function to the target domain, we realize that if the model has a low error on the source domain, it stops updating the feature extractor for future source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' C-DANN is a variant of the adversarial domain adaptation network in which we provide a representational memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This memory is used to store past data samples but has a limited size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' This emulates an incremental learning prob- lem setting where we store past samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We observe that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='HL一TR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='HRTR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='TLTR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='OLTR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='ORTR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Iteration100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='HLOL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='HR-OL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='TL-OL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='TROL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='OROL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Iteration100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='Accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='HL-OR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='HROR ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' acc Fgt(%) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' acc Fgt(%) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' acc Fgt(%) Hippocampal L 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='51% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='83% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='89% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='52% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='12% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='67% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='00% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='58% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='11% Table 2: A comparison of our method (ALeN) with unsupervised adversarial domain adaptation method (ADDA), unsuper- vised continual learning methods (CMA-GFK, CMA-SA), unsupervised multi-task learning methods (MDAN, B-DANN, C-DANN) and fine-tuning (FT) using only labeled source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' acc (%) is the average overall accuracy and Fgt(%) is the forgetting at the end of all iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We average the results over 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' the method could minimize forgetting using the stored sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' However, due to the limited size of the memory, for- getting occurs as the number of increments increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='3 Ablation study and sensitivity analysis We present a sensitivity analysis and an ablation study for the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We illustrate the effects of the anal- ysis on the automated disease diagnosis task for predict- ing Alzheimer’s disease using a non-stationary source do- main (simulated using regions Hippocampus Right, Tem- poral left and right, and occipital left and right) and non- stationary target domain data (sampled from Hippocampus Left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Effectiveness of Gaussian prototypes and OOD sam- ples: We analyzed the sensitivity of the hyper-parameter k used for carrying out negative sample identification in the foresighted learning algorithm (algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We use a 3-σ confidence interval to pseudo-label the |Cs + 1|th- class OOD samples following the empirical rule for Gaus- sian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In order to verify that our Gaussian esti- mates are accurate, we empirically tested the efficiency of the assumed confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We observed that 3-σ pro- vided the maximum predictive accuracy and best captured the source distribution characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' By setting k to 5, we can set the minimum number of negative samples, which can be considered an ablation of OOD samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We ob- serve the average accuracy for the target task to be 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='20% compared to 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='46% when using a 3-σ confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Effect of balancing source and target unlabelled data: We used a balanced source (Nsrc) and target (Nneg) do- main dataset to train our baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We test the ro- bustness of our model to imbalanced data by varying the Nsrc/Nneg ratio by ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Effect of class separation loss: We carry out the abla- tion study by removing the class separation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We learn the post-increment accuracy of the target domain classifier without applying the class separation loss (Ls1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We ob- serve that the average prediction accuracy without the loss minimization was 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='39% compared to 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content='46% using the class separation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' 5 CONCLUSION In this work, we present ALeN, an approach for unsuper- vised domain incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We present an empir- ical analysis of our approach by applying it domain incre- mental learning tasks in sentiment prediction and disease classification applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We compare our approach to existing state-of-the-art methods to show that the method achieves promising results all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We present a structured ablation study to observe the impact of dif- ferent components of our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Our work Abhinit Kumar Ambastha, Leong Tze Yun shows that we can incrementally update DNN classification models using unlabelled data without storing past training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' One of the limitations of this method is that it assumes the source and target data to belong to the same global fea- ture space, which might not be the case in several appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' We identify this problem setting as feature incre- mental learning, and aim to address it in our future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Also, due to limited size of the parametric space of the DNN models (assumed in our work), we observe that our method is prone to overfitting after multiple increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' In this paper, we have not analysed this issue and aim to do so in our future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Some works in the literature over- come this limitation by adding additional model parameters as new domains are incrementally added (Li and Hoiem, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' Polikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFLT4oBgHgl3EQfVy9V/content/2301.12054v1.pdf'} +page_content=' References J.' metadata={'source': 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b/aNAyT4oBgHgl3EQfifjx/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8576e1aac583c99260bd8185392142e5689831b44ce113aef03ea556788bf172 +size 1900589 diff --git a/atE4T4oBgHgl3EQfPAw4/content/tmp_files/2301.04969v1.pdf.txt b/atE4T4oBgHgl3EQfPAw4/content/tmp_files/2301.04969v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ddd37eec0c98bc4d542c8668c17f62885ce3c74 --- /dev/null +++ b/atE4T4oBgHgl3EQfPAw4/content/tmp_files/2301.04969v1.pdf.txt @@ -0,0 +1,1089 @@ +Ultra-low lattice thermal conductivity in Bi-doped KMgSb and role of spin-orbit +coupling in thermoelectric performance: An ab-initio study +Vikrant Chaudhary,1 Tashi Nautiyal,2 Tulika Maitra,2 Jeroen van den Brink,3, 4 and Hem C. Kandpal1, ∗ +1Indian Institute of Technology Roorkee, Department of Chemistry, Roorkee 247667, Uttarakhand, India +2Indian Institute of Technology Roorkee, Department of Physics, Roorkee 247667, Uttarakhand, India +3Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany +4Institute for Theoretical Physics and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Technische Universit¨at Dresden, 01069 Dresden, Germany +(Dated: January 13, 2023) +We systematically consider the KMgX family (X = P, As, and Sb) of materials and investigate the +effects of spin-orbit coupling (SOC) on thermal and electrical transport properties using a combined +first-principles and Boltzmann transport equation approach. +It is found that so far unexplored +ternary quasi-two-dimensional KMgSb is particularly promising with SOC having a strong effect, +shifting its behaviour from a p-type to n-type thermoelectric material. The transport properties +of KMgSb were studied under the application of hydrostatic pressure. Our calculations show that +Bi doping in KMgSb may prove to be a game changer as the lattice thermal conductivity (κL) +becomes ultra-low, thereby increasing the thermoelectric figure of merit (zT) by > 280 % with SOC +and by > 345 % without SOC. Through our computational investigation, we explain that the SOC +plays a critical role and establish that alloy engineering may improve thermoelectric performance +dramatically. +I. +INTRODUCTION +Thermometric materials have been around for many +decades and a plethora of materials have been developed +and identified for thermoelectric devices-based applica- +tions [1–9]. The performance and efficiency of thermo- +electric materials are judged using a dimensionless figure +of merit zT = S2σT/(κe + κL), where S, σ, κe, and +κL are Seebeck coefficient, electrical conductivity, elec- +tronic thermal conductivity, and lattice thermal conduc- +tivity, respectively. Among the many promising materi- +als [10–19], the materials that are layered in nature, like +PbTe, SnSe, show high zT value due to their low thermal +conductivity [2]. Thus, the quasi-2D nature of layered +materials leads to improved thermoelectric performance, +and such materials are widely used in thermoelectric de- +vices. +PbTe is one such material, which has been de- +veloped as a high-performance thermoelectric material +based exclusively on the structure and band engineering +[20]. With the same motive, we have studied the less ex- +plored KMgX (X = N, P, As, Sb, and Bi) family, where +the compounds have a quasi-2D type of crystal structure. +All the members of this family of compounds except for +KMgN have been experimentally realized [21]. +KMgBi was reported to be a narrow band gap (∼ +11.2 meV ) semiconductor [22] and has been reported +to be topologically non-trivial due to the presence of +type-I Dirac points [23]. +Contrary to this, a the den- +sity functional theory (DFT) based study has reported +that KMgBi has a larger band gap (∼ 280 meV ) and +predicts a high zT value (∼ 2.21) with p-type doping, +which is expected to improve further with alloy engineer- +∗ Corresponding author: hem.kandpal[at]cy.iitr.ac.in +ing [24]. +In the process of alloy engineering, one can +fortify the strength of SOC if heavier atoms replace the +lighter atoms. In such a case the inclusion of SOC be- +comes inescapable. In the current work, we are dealing +with Sb and Bi atoms and hence SOC has been given due +importance. +Besides KMgBi, a few studies are there on the other +members of KMgX (X = N, P, As, Sb, and Bi) fam- +ily in hexagonal, orthorhombic, and half-Heusler phases. +The thermoelectric properties of KMgP have been in- +vestigated in half-Heusler thin film [25] and MoS2 type +2D structure [26]. The ferroelectric and antiferroelectric +properties of KMgSb and KMgBi have been reported in +the hexagonal [27] and orthorhombic [28] phase, respec- +tively. In another study, the optical and elastic properties +of KMgX (X = N, P, As, Sb, and Bi) have been explored +in a hypothetical half-Heusler structure [29]. Apart from +these, no thermal and electronic transport studies have +been reported, as per our knowledge, in the above men- +tioned or experimentally realized tetragonal (P4/nmm) +structures. Thus, we have investigated the KMgX family +(X = P, As, Sb, and Bi) in the less studied tetragonal +structure[21]. We find that as we go down in the KMgX +family, there is band gap lowering, a desirable feature for +good thermoelectric performance. The studies on KMgBi +[23, 24] further motivated us to study KMgSb. Tuning of +the band gap and SOC strength of KMgSb by introduc- +ing Bi atoms in place of Sb makes an interesting subject +for this study. +The computational details and results for this work are +presented in Section II and III, respectively. Summary +and conclusions constitute Section IV. +arXiv:2301.04969v1 [cond-mat.mtrl-sci] 12 Jan 2023 + +2 +TABLE I. +The lattice parameter, band gap, and bulk modulus of the KMgX (X = P, As, Sb, and Bi)a in P4/nmm space +group. +XYZ +a (˚A) +c (˚A) +Band Gap Eg (eV) +B (GPa) +GGA +Expt. +GGA +Expt. +GGA +GGA+SOC +Expt. +KMgP +4.4601 +4.446 [21] +7.6499 +7.544 [21] +1.67 +1.65 +30.75 +KMgAs +4.5750 +4.546 [21] +7.8692 +7.716 [21] +1.12 +1.04 +27.36 +KMgSb +4.8380 +4.812 [21] +8.3488 +8.202 [21] +1.26 +1.07 +22.88 +KMgSb0.5Bi0.5 +4.9556 +NAb +8.7088 +NAb +0.58 +0.20 +16.07 +KMgBi +4.9338 +4.881 [21, 22] +8.5487 +8.382 [21, 22] +0.36 +0.0 +0.0112c [22] +19.85 +a KMgN has not been experimentally realized, hence not included in this work. +b NA: Not available +c Measured between 40 and 100 K. +II. +COMPUTATIONAL DETAILS +The computations performed in this work have three +major components, namely (a) DFT for electronic band +structure and density functional perturbation theory +(DFPT) for mechanical and dielectric properties, (b) +phonon or lattice dynamics, and (c) the scattering rates +and transport properties. +The DFT and DFPT calculation are done using the Vi- +enna ab-initio Simulation Package (VASP) [30–34]. Ini- +tially, the structural optimization was done using local +density approximation (LDA) and generalized gradient +approximation (GGA) of J. P. Perdew, K. Burke, and +M. Ernzerhof (PBE) [35] for the exchange-correlation +part. The GGA with SOC gives a band gap closer to +the experimental value in the case of KMgBi, therefore, +we have used the GGA-optimized structure for electronic +and thermal transport calculations. +All crystal struc- +tures have been optimized on a 21 × 21 × 21 k-mesh. +A cut-off energy of 550 eV for the wave function and +10−8 eV breaking condition on the convergence of the +self-consistency field (SCF) cycles were used. The ionic +relaxation was performed with a breaking condition of +0.001 eV/˚A on the forces. Furthermore, the mechanical +and dielectric properties were calculated using the DFPT +module of VASP and the results were used for scattering +rate evaluation. +The phonon band structures and Gr¨uneisen parame- +ters were obtained using the harmonic approximation in +the Phonopy package [36] and DFPT module of VASP. +For Gr¨uneisen parameter calculations, the volume of +the unit cell was changed by ∼ ±1.0%. +Further, the +lattice thermal conductivity (κL) was calculated using +Phono3py package [37] with VASP on a 5×5×3 k-mesh. +The last component in computations is related to the +scattering rates and transport properties, which are cal- +culated using AMSET code [38]. +AMSET takes in- +put like mechanical, piezoelectric, dielectric properties, +wave functions, band structure, and deformation poten- +tial from the VASP calculations. The scattering rates are +calculated using acoustic deformation potential (ADP), +ionized impurity (IMP), and polar optical phonon (POP) +scattering mechanisms. The ADP and IMP are elastic +processes, while POP is inelastic in nature [38]. In AM- +SET, scattering rates are calculated using Fermi’s golden +rule, +τ −1 +i→f = 2π +¯h |gfi(k, q)|2δ(ϵi − ϵf) +(1) +where i is the initial and f is the final state, τ is the +relaxation time, g is the coupling matrix, and ϵi (ϵf) is +the initial (final) energy of the electron. The relaxation +time obtained is automatically used in transport proper- +ties calculations by AMSET. +III. +RESULTS AND DISCUSSION +A. +Structural analysis and lattice dynamics +FIG. 1. The crystal structure of KMgX (X = P, As, Sb, and +Bi) in P4/nmm space group. +The compounds of KMgX family (X = P, As, Sb, and +Bi) crystallize in P4/nmm space group (No. 129) [21]. +The K, Mg, and X occupy 2c(0, 0.5, ∼ 0.65), 2a(0, 0, 0), +and 2c(0, 0.5, ∼ 0.20) sites, respectively. The z compo- +nent of the sites occupied by K and X are free positions, +which we have relaxed in structural optimization. The + +X +Mg +K3 +400 +600 +800 +T (K) +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +κL (W m +-1 K +-1 ) +400 +600 +800 +T (K) +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +x, y +z +Avg. +400 +600 +800 +T (K) +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +400 +600 +800 +T (K) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(a) +(c) +(b) +(d) +FIG. 2. The lattice thermal conductivity of KMgSb at (a) -1.0 GPa, (b) ambient, (c) 2.0 GPa pressure. (d) The κL of a +K-terminated freely suspended layer of KMgSb, where heat can only travel in a-b plane. +structures of KMgX are layered in the c direction, con- +sisting of alternating K and Mg-X layers. In the Mg-X +layer, Mg and X atoms form edge-sharing tetrahedra, as +shown in Fig. 1. Since K is an alkali metal, it tends to +give an electron to the MgX tetrahedron, resulting in an +electrostatic interaction between K+ and [MgX]−. This +electrostatic interaction is weaker than the covalent bond +between Mg and X atoms due to the large separation be- +tween K and Mg-X layers. +As a result of the layered +nature, KMgX can be easily cleaved along the a-b plane. +Furthermore, KMgX family compounds have a lower +bulk modulus than the popular layered thermoelectric +materials PbTe and PbSe [39], indicating their low +strength. +In Table I, the optimized lattice constants, +band gap (Eg), and bulk modulus of KMgX members are +listed. As we go down in the table, the strength of SOC +increases due to the increasing atomic number of the X +atom. We note that KMgP is less relevant for the current +study due to a large band gap (Eg ∼ 1.67). The KMgAs +and KMgSb have a closely matching Eg (∼ 1.0 eV ) but +utterly different transport properties (as we found out +in calculations), probably due to the larger SOC for the +latter. The transport properties of KMgAs when investi- +gated gave no exciting results. Now onwards, the trans- +port properties of KMgSb have been discussed, starting +with the lattice dynamics. +The phonon band structure and mode Gr¨uneisen pa- +rameter (γ), the key tools to understand the various +aspects of lattice dynamics, were calculated using the +phonopy package [36]. The mode Gr¨uneisen parameter +is a direct measure of the degree of anharmonicity in +phonon modes. A larger value of γ leads to a low κL, as +per the inverse square relation (κL ∝ 1/γ2) [40]. +To check the stability of KMgSb under pressure varia- +tion, the effect of positive and negative pressure on its +transport properties has been investigated thoroughly. +The maximum value of γ (∼ 2.05 at ambient pressure) +rises to ∼ 4.1 at -1.0 GPa indicating an encouraging drop +in κL [41]. Accordingly, Fig. 2(a) and (c) show that the +average and directional κL decrease under the applica- +tion of negative pressure. This reduction is purely due +to the change in the bonding strength when the volume +of the cell increases. +As discussed above, the KMgSb +can be easily cleaved, and this motivated us to check the +κL of a freely suspended K-terminated layer also by us- +ing a supercell approach. We observed that at 300 K +the κL in x-y plane reduces to ∼ +2.3 Wm−1K−1 (in +freely suspended layer) from ∼ 3.7 Wm−1K−1 in bulk +(or quasi-2D) KMgSb, as seen from Fig. 2(b) and (d). +This indicates that improved thermoelectric performance +may be achieved in the freely suspended layer (or mono- +layer) of KMgSb. +Considering that the κL decreases as we go towards +negative pressure, it is worth exploring further. The neg- +ative pressure may seem to be impractical, but it may be +mimicked by chemically doping bigger size elements in +the unit cell, thereby increasing the cell volume. +In a +recent computational study, KMgBi was reported as an +excellent thermoelectric material with a desirably high +zT value (> 2.0) [22]. KMgSb being closer to KMgBi in +size (than e. g. KMgAs), makes KMgSb an interesting +candidate for experimentalists who can replace Sb with +Bi atoms and fine-tune the thermal and electronic trans- +port properties. We designed a composition by replacing +50 % Sb atoms with Bi atoms using the supercell ap- +proach. The substitution is done in such a way that the +resulting material KMgSb0.5Bi0.5 remains in the highest +possible space group. +The phonon band structure of KMgSb0.5Bi0.5 contains +72 bands, where the three lowest frequency bands (color +bands in Fig. +3(a)) originating from Γ point are the +acoustic bands. +In Fig. +3(d), the cumulative lattice +thermal conductivity (κc) clearly shows that the low- +frequency acoustic modes carry most of the heat in the +crystal. All three acoustic modes, two transverse acous- +tic (TA) and one longitudinal acoustic (LA), are seen up +to a frequency of ∼ 1.25 THz. +The change in mode Gr¨uneisen parameter after Bi dop- +ing is observed in the low-frequency modes, which con- +tribute more than 82% in κL as shown in Fig. +3(d). +The maximum value of γ increases to ∼ 15 (see Fig. +3(b)) in the low-frequency region; this increased value is + +4 +TABLE II. The longitudinal (vL) and transverse (vT ) speed of sound, minimum lattice thermal conductivity from Cahill model +(κglass), and diffusive thermal conductivity (κdiff) of KMgSb0.5Bi0.5. +Direction +v1 +T +v2 +T +vL +κglass[42, 43] +κdiff[44] +(ms−1) +(ms−1) +(ms−1) +(Wm−1K−1) +(Wm−1K−1) +Γ → X +1999.12 +1646.02 +3302.90 +0.36 +0.22 +Γ → M +1969.71 +1625.56 +3221.70 +0.35 +0.22 +Γ → Z +1626.87 +1626.87 +2125.68 +0.28 +0.17 +Γ X M +Γ Z R A +0.0 +2.0 +4.0 +6.0 +8.0 +Frequency (THz) +0 +2 +4 +6 +8 +Frequency (THz) +0.0 +6.0 +12.0 +18.0 +γ +400 +600 +800 +T (K) +0.0 +0.2 +0.4 +0.6 +0.8 +κL (W m +-1 K +-1) +X, Y +Z +Avg. +0 +2 +4 +6 +8 +Frequency (THz) +0.0 +0.2 +0.4 +0.6 +0.8 +κc (W m +-1 K +-1) +(a) +(b) +(c) +(d) +700 +FIG. 3. (a) Phonon band structure, (b) Gr¨uneisen parameter, +(c) lattice thermal conductivity (κL), and (d) cumulative lat- +tice thermal (κc) conductivity (at 300 K) of KMgSb0.5Bi0.5. +The dotted line in (d) represents the derivative of κc. +≈ 7.5 times larger than the γ value of both, KMgSb and +KMgBi [24]. Moreover, the mode Gr¨uneisen parameter of +KMgSb0.5Bi0.5 is even larger than that for the popularly +used thermoelectric material quasi-2D SnSe (γ ∼ 7.2) +[45] and other alkali metal based compounds like K3Sb +(γ ∼ 4.0), Rb3Sb (γ ∼ 3.5), and Cs3Sb (γ ∼ 2.4) [46]. +This increased value of γ is clearly an indication of a +decrease in the lattice thermal conductivity. +Noteworthily, a very large change in κL is seen (Fig. +3(c)), with respect to the pristine KMgSb (Fig. 2(b)) +on replacing 50 % Sb by Bi atoms. This ultra-low κL +of KMgSb0.5Bi0.5 is in accordance with the large mode +Gr¨uneisen parameter in low-frequency region (Fig. 3 (b)) +and low speed of sound (vT < 2000 ms−1) obtained for +acoustic modes near the Γ point (Table II). This dramatic +change in κ can be associated with (a) the increased vol- +ume of the cell and (b) the strain induced in the lattice +due to the substitutional defects created after Bi doping. +The ultra-low κL, together with the quasi-2D struc- +ture and low bulk modulus of KMgSb0.5Bi0.5, prompted +us to check the amorphous limit of the lattice thermal +conductivity. +The minimum value of κL can be esti- +mated using κglass = 1.21n2/3kB(2vT + vL)/3 [42, 43] +and κdiff ≈ 0.76n2/3kB(2vT + vL)/3 [44], where n, kB, +vT , and vL are the number density of atoms, Boltzmann +constant, transverse, and longitudinal speed of sound, re- +spectively. The κglass is a good approximation for disor- +dered and amorphous solids [47], whereas the κdiff gives +a better approximation for crystalline solids. We have +used κdiff as a predictor of minimum possible κL in our +work. +As listed in Table II, the κdiff in Γ → X and +Γ → M directions are equal, and are greater than the +value obtained in Γ → Z direction, which is a direct +consequence of the quasi-2D nature of this compound. +The limiting value of κdiff is reached at ∼ 400 K in +cross-plane, whereas in-plane the κL is higher than the +κdiff even up to 800 K. On an average, the minimum +possible κL is obtained to be ∼ 0.19 at 700 K temper- +ature. Therefore, we have investigated ahead the trans- +port properties of KMgSb0.5Bi0.5 between 300 K (room +temperature) and 700 K. To be pragmatic, we have pre- +sented and discussed the electronic and thermal transport +properties only at 300 K and 600 K. +B. +Electronic and Thermal transport +The electronic and thermal transport properties are +obtained by numerically solving the linearized Boltzmann +transport equations (BTEs). +Various transport coeffi- +cients are obtained from the generalized transport equa- +tion +Lα(µ, T) = q2 +� +Σ(ϵ) (ϵ − µ)α +� +−∂f 0(ϵ, T) +∂ϵ +� +dϵ +(2) +where, q, Σ(ϵ), µ and f 0(ϵ, T) are electronic charge, spec- +tral conductivity, chemical potential, and Fermi-Dirac +distribution function, respectively [38, 48, 49]. The See- +beck coefficient (S), electronic conductivity (σ), and elec- +tronic component of thermal conductivity (κe), are given +by +1 +qT +L1 +L0 , L0, and +1 +q2T +� +(L1)2 +L0 +− L2� +, respectively. For a +very long time, these BTEs were solved within the con- +stant relaxation time approach (CRTA) due to the com- +plex nature of the scattering processes. The electronic +relaxation time can be calculated using the AMSET [38] +package based on Fermi’s golden rule and the scattering +mechanisms discussed in Section II. +Having discussed the lattice thermal conductivity in +sub-section III A, we now focus on thermoelectric trans- +port. +At ambient pressure, maximum zT (∼ 0.4) for +KMgSb is obtained without SOC in the p-type region at + +5 +10 +19 +10 +20 +10 +21 +carrier concentration (cm +-3) +0.0 +0.4 +0.8 +1.2 +1.6 +S +2σ (mW m +-1 K +-2) +300K +600K +10 +19 +10 +20 +10 +21 +10 +19 +10 +20 +10 +21 +carrier concentration (cm +-3) +0.0 +0.4 +0.8 +1.2 +1.6 +2.0 +κe (W m +-1K +-1) +10 +19 +10 +20 +10 +21 +10 +19 +10 +20 +10 +21 +carrier concentration (cm +-3) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +σ (10 +5 S m +-1) +10 +19 +10 +20 +10 +21 +10 +19 +10 +20 +10 +21 +carrier concentration (cm +-3) +0.0 +0.5 +1.0 +1.5 +2.0 +zT +10 +19 +10 +20 +10 +21 +(a) +(b) +(c) +(d) +n-type +p-type +n-type +p-type +n-type +p-type +n-type +p-type +FIG. 4. (a) The Seebeck coefficient (S), (b) electronic component of thermal conductivity (κe), (c) electrical conductivity (σ), +and (d) thermoelectric figure of merit (zT) of KMgSb0.5Bi0.5 with (dotted lines) and without (solid lines) spin-orbit coupling. +600 K. The SOC reduces this zT value significantly to +∼ 0.13 in the p-type region, whereas only minor changes +are observed in the n-type region after including SOC. As +a result, the n-type behaviour is expected to dominate in +comparison to the p-type behaviour. This and the rest of +the transport properties of KMgSb have been discussed +in detail in the Supplemental Material, Fig. S2 [41]. +In Fig. +5, the band structure of KMgSb0.5Bi0.5 is +shown, where flat (in Γ → Z direction and valance re- +gion) as well as large dispersion (in conduction region) +bands are seen. +The band gap of KMgSb0.5Bi0.5 is +obtained as ∼ 0.58 eV without SOC, which drops to +∼ 0.20 eV with SOC (Table I). Bi doping has signifi- +cantly reduced the band gap. Such tunability of Eg (with +Γ X M Γ Z R A +-2 +-1 +0 +1 +2 +E (eV) +Γ X M Γ Z R A +-2 +-1 +0 +1 +2 +(a) +(b) +FIG. 5. The electronic band structure of KMgSb0.5Bi0.5 (a) +without and (b) with spin-orbit coupling. +doping) is highly desirable while designing materials for +thermoelectric applications. In Fig. 4, we have presented +the power factor (PF), κe, σ, and zT of KMgSb0.5Bi0.5. +The zT value of KMgSb and KMgSb0.5Bi0.5 increases +with temperature. +At 600 K, the peak zT value of +KMgSb0.5Bi0.5 is ∼ 1.75 (∼ 0.51) without (with) SOC, +which is obtained in the p-type region as shown in Fig. +4(d). So, even though KMgSb changes from p-type to n- +type thermoelectric material (see Supplemental Material, +Fig. S2 [41]), KMgSb0.5Bi0.5 remains a p-type material +even after including SOC in the calculations. The PF and +κe of KMgSb (see Supplemental Material, Fig. S2 [41]) +and KMgSb0.5Bi0.5 at 600 K are closely matching at the +doping level of peak zT value, which indicates that the +thermoelectric performance enhancement is purely due +to the reduced κL after alloy engineering. If we compare +the peak zT value of KMgSb and KMgSb0.5Bi0.5 in the +p-type region, an increase of ∼ 346.14% and 282.14% is +achieved without and with SOC after alloy engineering. +Although the SOC affects the zT value adversely, alloy +engineering seems to be a promising route to induce neg- +ative pressure and thus improve the zT dramatically. +One important aspect of thermoelectric transport is +the electronic relaxation time, which is a complex phe- +nomenon. As discussed above, the CRTA was considered +to be the most convenient and helpful choice in electronic +transport calculations, but it comes with a cost. Under +CRTA, the Seebeck and Hall coefficients become inde- +pendent of scattering rates [50]. Moreover, a constant +value of τ cannot account for all the scattering processes +involved in electronic transport. To get an insight, we + +6 +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +ne = 1019 cm +3 T = 300 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +ne = 1019 cm +3 T = 600 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +ne = 1020 cm +3 T = 300 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +ne = 1020 cm +3 T = 600 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +nh = 1019 cm +3 T = 300 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +nh = 1019 cm +3 T = 600 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +nh = 1020 cm +3 T = 300 K +ADP +IMP +POP +total +1.0 +0.5 +0.0 +0.5 +1.0 + (eV) +10 +18 +10 +16 +10 +14 +10 +12 +10 +10 + (s) +nh = 1020 cm +3 T = 600 K +ADP +IMP +POP +total +1 +FIG. 6. The scattering mechanism dependent electronic relaxation time (τ) in KMgSb0.5Bi0.5, calculated at 300 K and 600 K +for 1019 cm−3 and 1020 cm−3 carrier concentration. +have shown the scattering mechanism dependent relax- +ation time (τ) in Fig. 6. The overall electronic relaxation +time is found to lie between 10−14 s and 10−16 s, which +arises primarily from the POP with a small contribution +from ADP and IMP scattering mechanisms. The average +value of τ is found to be larger for n-type doping than +for p-type. +The results discussed above explain the impact of pres- +sure, SOC, and alloy engineering on thermal and elec- +tronic transport. We observed a quantitative as well as a +qualitative difference between the thermoelectric trans- +port results obtained with and without SOC. Moreover, +the thermoelectric performance shows a huge improve- +ment after Bi doping. Our results suggest that the ther- +moelectric figure of merit (zT) should not be labelled +merely by a single number. Therefore, we propose that +the maximum zT value of KMgSb0.5Bi0.5 at 600 K may +be between 0.51 and 1.75, depending upon the SOC +strength and carrier concentration obtained in an exper- +imentally grown sample. +IV. +SUMMARY AND CONCLUSIONS +In this work, we started with the investigation of the +thermal and electronic transport properties of the KMgX +family (X = P, As, and Sb). The large SOC strength and +∼ 1.0 eV band gap prompted us to zero in on KMgSb for +this study. We find that in KMgSb the thermoelectric +transport is strongly dependent on the SOC. The peak +zT value of KMgSb without SOC (at 600 K) is ∼ 0.38 +in the p-type region; after the inclusion of SOC the peak +shifts to ∼ 0.24 in the n-type region. +It is encouraging to note that the κL of KMgSb de- +creases under negative pressure, implying the applica- +tion of negative pressure, e. g. by doping with larger size +atoms, would be helpful in improving the thermoelectric +performance. This finding guided us to explore a new ma- +terial by substitutionally doping Bi (in place of smaller +size Sb) in KMgSb. The alloy engineering does improve +thermoelectric performance dramatically. On 50% sub- +stitutional doping of Bi in place of Sb atoms, the κL be- +comes ultra-low and the zT without (with) SOC changes +from ∼ 0.39 (∼ 0.14) to ∼ 1.75 (∼ 0.51) in the p-type +region. +To conclude, KMgSb0.5Bi0.5 is a potential candidate +for thermal barrier applications due to its ultra-low κL. +Even though the SOC deteriorates the thermoelectric +performance (in comparison to the results without SOC) +and cannot be done away with in the materials contain- +ing heavy elements, alloy engineering may come as a big +respite as it can improve the thermoelectric performance +dramatically. +ACKNOWLEDGMENTS +This work used the Supercomputing facility of IIT +Roorkee established under National Supercomputing +Mission (NSM), Government of India and supported +by Centre for Development of Advanced Computing +(CDAC), Pune. We have also used other computational +facilities provided by Institute Computer Center (ICC), +IIT Roorkee. 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B 56, R1650 +(1997). + diff --git a/atE4T4oBgHgl3EQfPAw4/content/tmp_files/load_file.txt b/atE4T4oBgHgl3EQfPAw4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9b69ed4e6b64d8d8f1edeaaf58e6378f3858036 --- /dev/null +++ b/atE4T4oBgHgl3EQfPAw4/content/tmp_files/load_file.txt @@ -0,0 +1,943 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf,len=942 +page_content='Ultra-low lattice thermal conductivity in Bi-doped KMgSb and role of spin-orbit coupling in thermoelectric performance: An ab-initio study Vikrant Chaudhary,1 Tashi Nautiyal,2 Tulika Maitra,2 Jeroen van den Brink,3, 4 and Hem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Kandpal1, ∗ 1Indian Institute of Technology Roorkee, Department of Chemistry, Roorkee 247667, Uttarakhand, India 2Indian Institute of Technology Roorkee, Department of Physics, Roorkee 247667, Uttarakhand, India 3Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany 4Institute for Theoretical Physics and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='qmat, Technische Universit¨at Dresden, 01069 Dresden, Germany (Dated: January 13, 2023) We systematically consider the KMgX family (X = P, As, and Sb) of materials and investigate the effects of spin-orbit coupling (SOC) on thermal and electrical transport properties using a combined first-principles and Boltzmann transport equation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' It is found that so far unexplored ternary quasi-two-dimensional KMgSb is particularly promising with SOC having a strong effect, shifting its behaviour from a p-type to n-type thermoelectric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The transport properties of KMgSb were studied under the application of hydrostatic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Our calculations show that Bi doping in KMgSb may prove to be a game changer as the lattice thermal conductivity (κL) becomes ultra-low, thereby increasing the thermoelectric figure of merit (zT) by > 280 % with SOC and by > 345 % without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Through our computational investigation, we explain that the SOC plays a critical role and establish that alloy engineering may improve thermoelectric performance dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' INTRODUCTION Thermometric materials have been around for many decades and a plethora of materials have been developed and identified for thermoelectric devices-based applica- tions [1–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The performance and efficiency of thermo- electric materials are judged using a dimensionless figure of merit zT = S2σT/(κe + κL), where S, σ, κe, and κL are Seebeck coefficient, electrical conductivity, elec- tronic thermal conductivity, and lattice thermal conduc- tivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Among the many promising materi- als [10–19], the materials that are layered in nature, like PbTe, SnSe, show high zT value due to their low thermal conductivity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Thus, the quasi-2D nature of layered materials leads to improved thermoelectric performance, and such materials are widely used in thermoelectric de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' PbTe is one such material, which has been de- veloped as a high-performance thermoelectric material based exclusively on the structure and band engineering [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' With the same motive, we have studied the less ex- plored KMgX (X = N, P, As, Sb, and Bi) family, where the compounds have a quasi-2D type of crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' All the members of this family of compounds except for KMgN have been experimentally realized [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' KMgBi was reported to be a narrow band gap (∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='2 meV ) semiconductor [22] and has been reported to be topologically non-trivial due to the presence of type-I Dirac points [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Contrary to this, a the den- sity functional theory (DFT) based study has reported that KMgBi has a larger band gap (∼ 280 meV ) and predicts a high zT value (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='21) with p-type doping, which is expected to improve further with alloy engineer- ∗ Corresponding author: hem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='kandpal[at]cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='in ing [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In the process of alloy engineering, one can fortify the strength of SOC if heavier atoms replace the lighter atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In such a case the inclusion of SOC be- comes inescapable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In the current work, we are dealing with Sb and Bi atoms and hence SOC has been given due importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Besides KMgBi, a few studies are there on the other members of KMgX (X = N, P, As, Sb, and Bi) fam- ily in hexagonal, orthorhombic, and half-Heusler phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The thermoelectric properties of KMgP have been in- vestigated in half-Heusler thin film [25] and MoS2 type 2D structure [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The ferroelectric and antiferroelectric properties of KMgSb and KMgBi have been reported in the hexagonal [27] and orthorhombic [28] phase, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In another study, the optical and elastic properties of KMgX (X = N, P, As, Sb, and Bi) have been explored in a hypothetical half-Heusler structure [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Apart from these, no thermal and electronic transport studies have been reported, as per our knowledge, in the above men- tioned or experimentally realized tetragonal (P4/nmm) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Thus, we have investigated the KMgX family (X = P, As, Sb, and Bi) in the less studied tetragonal structure[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We find that as we go down in the KMgX family, there is band gap lowering, a desirable feature for good thermoelectric performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The studies on KMgBi [23, 24] further motivated us to study KMgSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Tuning of the band gap and SOC strength of KMgSb by introduc- ing Bi atoms in place of Sb makes an interesting subject for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The computational details and results for this work are presented in Section II and III, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Summary and conclusions constitute Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='04969v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The lattice parameter, band gap, and bulk modulus of the KMgX (X = P, As, Sb, and Bi)a in P4/nmm space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' XYZ a (˚A) c (˚A) Band Gap Eg (eV) B (GPa) GGA Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' GGA Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' GGA GGA+SOC Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' KMgP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4601 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='446 [21] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='6499 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='544 [21] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='65 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='75 KMgAs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5750 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='546 [21] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='8692 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='716 [21] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='04 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='36 KMgSb 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='8380 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='812 [21] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='3488 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='202 [21] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='07 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='88 KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='9556 NAb 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='7088 NAb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='20 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='07 KMgBi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='9338 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='881 [21, 22] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5487 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='382 [21, 22] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0112c [22] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='85 a KMgN has not been experimentally realized, hence not included in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' b NA: Not available c Measured between 40 and 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' COMPUTATIONAL DETAILS The computations performed in this work have three major components, namely (a) DFT for electronic band structure and density functional perturbation theory (DFPT) for mechanical and dielectric properties, (b) phonon or lattice dynamics, and (c) the scattering rates and transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The DFT and DFPT calculation are done using the Vi- enna ab-initio Simulation Package (VASP) [30–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Ini- tially, the structural optimization was done using local density approximation (LDA) and generalized gradient approximation (GGA) of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Perdew, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Burke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Ernzerhof (PBE) [35] for the exchange-correlation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The GGA with SOC gives a band gap closer to the experimental value in the case of KMgBi, therefore, we have used the GGA-optimized structure for electronic and thermal transport calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' All crystal struc- tures have been optimized on a 21 × 21 × 21 k-mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' A cut-off energy of 550 eV for the wave function and 10−8 eV breaking condition on the convergence of the self-consistency field (SCF) cycles were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The ionic relaxation was performed with a breaking condition of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='001 eV/˚A on the forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Furthermore, the mechanical and dielectric properties were calculated using the DFPT module of VASP and the results were used for scattering rate evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The phonon band structures and Gr¨uneisen parame- ters were obtained using the harmonic approximation in the Phonopy package [36] and DFPT module of VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' For Gr¨uneisen parameter calculations, the volume of the unit cell was changed by ∼ ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Further, the lattice thermal conductivity (κL) was calculated using Phono3py package [37] with VASP on a 5×5×3 k-mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The last component in computations is related to the scattering rates and transport properties, which are cal- culated using AMSET code [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' AMSET takes in- put like mechanical, piezoelectric, dielectric properties, wave functions, band structure, and deformation poten- tial from the VASP calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The scattering rates are calculated using acoustic deformation potential (ADP), ionized impurity (IMP), and polar optical phonon (POP) scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The ADP and IMP are elastic processes, while POP is inelastic in nature [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In AM- SET, scattering rates are calculated using Fermi’s golden rule, τ −1 i→f = 2π ¯h |gfi(k, q)|2δ(ϵi − ϵf) (1) where i is the initial and f is the final state, τ is the relaxation time, g is the coupling matrix, and ϵi (ϵf) is the initial (final) energy of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The relaxation time obtained is automatically used in transport proper- ties calculations by AMSET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Structural analysis and lattice dynamics FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The crystal structure of KMgX (X = P, As, Sb, and Bi) in P4/nmm space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The compounds of KMgX family (X = P, As, Sb, and Bi) crystallize in P4/nmm space group (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 129) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The K, Mg, and X occupy 2c(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='65), 2a(0, 0, 0), and 2c(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='20) sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The z compo- nent of the sites occupied by K and X are free positions, which we have relaxed in structural optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The X Mg K3 400 600 800 T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 κL (W m 1 K 1 ) 400 600 800 T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 x, y z Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 400 600 800 T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 400 600 800 T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (a) (c) (b) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The lattice thermal conductivity of KMgSb at (a) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 GPa, (b) ambient, (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 GPa pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' (d) The κL of a K-terminated freely suspended layer of KMgSb, where heat can only travel in a-b plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' structures of KMgX are layered in the c direction, con- sisting of alternating K and Mg-X layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In the Mg-X layer, Mg and X atoms form edge-sharing tetrahedra, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Since K is an alkali metal, it tends to give an electron to the MgX tetrahedron, resulting in an electrostatic interaction between K+ and [MgX]−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This electrostatic interaction is weaker than the covalent bond between Mg and X atoms due to the large separation be- tween K and Mg-X layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' As a result of the layered nature, KMgX can be easily cleaved along the a-b plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Furthermore, KMgX family compounds have a lower bulk modulus than the popular layered thermoelectric materials PbTe and PbSe [39], indicating their low strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In Table I, the optimized lattice constants, band gap (Eg), and bulk modulus of KMgX members are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' As we go down in the table, the strength of SOC increases due to the increasing atomic number of the X atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We note that KMgP is less relevant for the current study due to a large band gap (Eg ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The KMgAs and KMgSb have a closely matching Eg (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 eV ) but utterly different transport properties (as we found out in calculations), probably due to the larger SOC for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The transport properties of KMgAs when investi- gated gave no exciting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Now onwards, the trans- port properties of KMgSb have been discussed, starting with the lattice dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The phonon band structure and mode Gr¨uneisen pa- rameter (γ), the key tools to understand the various aspects of lattice dynamics, were calculated using the phonopy package [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The mode Gr¨uneisen parameter is a direct measure of the degree of anharmonicity in phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' A larger value of γ leads to a low κL, as per the inverse square relation (κL ∝ 1/γ2) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' To check the stability of KMgSb under pressure varia- tion, the effect of positive and negative pressure on its transport properties has been investigated thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The maximum value of γ (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='05 at ambient pressure) rises to ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='1 at -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 GPa indicating an encouraging drop in κL [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Accordingly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 2(a) and (c) show that the average and directional κL decrease under the applica- tion of negative pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This reduction is purely due to the change in the bonding strength when the volume of the cell increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' As discussed above, the KMgSb can be easily cleaved, and this motivated us to check the κL of a freely suspended K-terminated layer also by us- ing a supercell approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We observed that at 300 K the κL in x-y plane reduces to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='3 Wm−1K−1 (in freely suspended layer) from ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='7 Wm−1K−1 in bulk (or quasi-2D) KMgSb, as seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 2(b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This indicates that improved thermoelectric performance may be achieved in the freely suspended layer (or mono- layer) of KMgSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Considering that the κL decreases as we go towards negative pressure, it is worth exploring further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The neg- ative pressure may seem to be impractical, but it may be mimicked by chemically doping bigger size elements in the unit cell, thereby increasing the cell volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In a recent computational study, KMgBi was reported as an excellent thermoelectric material with a desirably high zT value (> 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' KMgSb being closer to KMgBi in size (than e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' KMgAs), makes KMgSb an interesting candidate for experimentalists who can replace Sb with Bi atoms and fine-tune the thermal and electronic trans- port properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We designed a composition by replacing 50 % Sb atoms with Bi atoms using the supercell ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The substitution is done in such a way that the resulting material KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 remains in the highest possible space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The phonon band structure of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 contains 72 bands, where the three lowest frequency bands (color bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3(a)) originating from Γ point are the acoustic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3(d), the cumulative lattice thermal conductivity (κc) clearly shows that the low- frequency acoustic modes carry most of the heat in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' All three acoustic modes, two transverse acous- tic (TA) and one longitudinal acoustic (LA), are seen up to a frequency of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='25 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The change in mode Gr¨uneisen parameter after Bi dop- ing is observed in the low-frequency modes, which con- tribute more than 82% in κL as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The maximum value of γ increases to ∼ 15 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3(b)) in the low-frequency region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' this increased value is 4 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The longitudinal (vL) and transverse (vT ) speed of sound, minimum lattice thermal conductivity from Cahill model (κglass), and diffusive thermal conductivity (κdiff) of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Direction v1 T v2 T vL κglass[42, 43] κdiff[44] (ms−1) (ms−1) (ms−1) (Wm−1K−1) (Wm−1K−1) Γ → X 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='12 1646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='02 3302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='22 Γ → M 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='71 1625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='56 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='22 Γ → Z 1626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='87 1626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='87 2125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='17 Γ X M Γ Z R A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 Frequency (THz) 0 2 4 6 8 Frequency (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 γ 400 600 800 T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='8 κL (W m 1 K 1) X, Y Z Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 0 2 4 6 8 Frequency (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='8 κc (W m 1 K 1) (a) (b) (c) (d) 700 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' (a) Phonon band structure, (b) Gr¨uneisen parameter, (c) lattice thermal conductivity (κL), and (d) cumulative lat- tice thermal (κc) conductivity (at 300 K) of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The dotted line in (d) represents the derivative of κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 times larger than the γ value of both, KMgSb and KMgBi [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Moreover, the mode Gr¨uneisen parameter of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 is even larger than that for the popularly used thermoelectric material quasi-2D SnSe (γ ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='2) [45] and other alkali metal based compounds like K3Sb (γ ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0), Rb3Sb (γ ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5), and Cs3Sb (γ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This increased value of γ is clearly an indication of a decrease in the lattice thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Noteworthily, a very large change in κL is seen (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3(c)), with respect to the pristine KMgSb (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 2(b)) on replacing 50 % Sb by Bi atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This ultra-low κL of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 is in accordance with the large mode Gr¨uneisen parameter in low-frequency region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3 (b)) and low speed of sound (vT < 2000 ms−1) obtained for acoustic modes near the Γ point (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This dramatic change in κ can be associated with (a) the increased vol- ume of the cell and (b) the strain induced in the lattice due to the substitutional defects created after Bi doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The ultra-low κL, together with the quasi-2D struc- ture and low bulk modulus of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5, prompted us to check the amorphous limit of the lattice thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The minimum value of κL can be esti- mated using κglass = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='21n2/3kB(2vT + vL)/3 [42, 43] and κdiff ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='76n2/3kB(2vT + vL)/3 [44], where n, kB, vT , and vL are the number density of atoms, Boltzmann constant, transverse, and longitudinal speed of sound, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The κglass is a good approximation for disor- dered and amorphous solids [47], whereas the κdiff gives a better approximation for crystalline solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We have used κdiff as a predictor of minimum possible κL in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' As listed in Table II, the κdiff in Γ → X and Γ → M directions are equal, and are greater than the value obtained in Γ → Z direction, which is a direct consequence of the quasi-2D nature of this compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The limiting value of κdiff is reached at ∼ 400 K in cross-plane, whereas in-plane the κL is higher than the κdiff even up to 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' On an average, the minimum possible κL is obtained to be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='19 at 700 K temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Therefore, we have investigated ahead the trans- port properties of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 between 300 K (room temperature) and 700 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' To be pragmatic, we have pre- sented and discussed the electronic and thermal transport properties only at 300 K and 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Electronic and Thermal transport The electronic and thermal transport properties are obtained by numerically solving the linearized Boltzmann transport equations (BTEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Various transport coeffi- cients are obtained from the generalized transport equa- tion Lα(µ, T) = q2 � Σ(ϵ) (ϵ − µ)α � −∂f 0(ϵ, T) ∂ϵ � dϵ (2) where, q, Σ(ϵ), µ and f 0(ϵ, T) are electronic charge, spec- tral conductivity, chemical potential, and Fermi-Dirac distribution function, respectively [38, 48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The See- beck coefficient (S), electronic conductivity (σ), and elec- tronic component of thermal conductivity (κe), are given by 1 qT L1 L0 , L0, and 1 q2T � (L1)2 L0 − L2� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' For a very long time, these BTEs were solved within the con- stant relaxation time approach (CRTA) due to the com- plex nature of the scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The electronic relaxation time can be calculated using the AMSET [38] package based on Fermi’s golden rule and the scattering mechanisms discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Having discussed the lattice thermal conductivity in sub-section III A, we now focus on thermoelectric trans- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' At ambient pressure, maximum zT (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4) for KMgSb is obtained without SOC in the p-type region at 5 10 19 10 20 10 21 carrier concentration (cm 3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='6 S 2σ (mW m 1 K 2) 300K 600K 10 19 10 20 10 21 10 19 10 20 10 21 carrier concentration (cm 3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 κe (W m 1K 1) 10 19 10 20 10 21 10 19 10 20 10 21 carrier concentration (cm 3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 σ (10 5 S m 1) 10 19 10 20 10 21 10 19 10 20 10 21 carrier concentration (cm 3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 zT 10 19 10 20 10 21 (a) (b) (c) (d) n-type p-type n-type p-type n-type p-type n-type p-type FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' (a) The Seebeck coefficient (S), (b) electronic component of thermal conductivity (κe), (c) electrical conductivity (σ), and (d) thermoelectric figure of merit (zT) of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 with (dotted lines) and without (solid lines) spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The SOC reduces this zT value significantly to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='13 in the p-type region, whereas only minor changes are observed in the n-type region after including SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' As a result, the n-type behaviour is expected to dominate in comparison to the p-type behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This and the rest of the transport properties of KMgSb have been discussed in detail in the Supplemental Material, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' S2 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 5, the band structure of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 is shown, where flat (in Γ → Z direction and valance re- gion) as well as large dispersion (in conduction region) bands are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The band gap of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 is obtained as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='58 eV without SOC, which drops to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='20 eV with SOC (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Bi doping has signifi- cantly reduced the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Such tunability of Eg (with Γ X M Γ Z R A 2 1 0 1 2 E (eV) Γ X M Γ Z R A 2 1 0 1 2 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The electronic band structure of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 (a) without and (b) with spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' doping) is highly desirable while designing materials for thermoelectric applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 4, we have presented the power factor (PF), κe, σ, and zT of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The zT value of KMgSb and KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 increases with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' At 600 K, the peak zT value of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='75 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='51) without (with) SOC, which is obtained in the p-type region as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' So, even though KMgSb changes from p-type to n- type thermoelectric material (see Supplemental Material, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' S2 [41]), KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 remains a p-type material even after including SOC in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The PF and κe of KMgSb (see Supplemental Material, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' S2 [41]) and KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 at 600 K are closely matching at the doping level of peak zT value, which indicates that the thermoelectric performance enhancement is purely due to the reduced κL after alloy engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' If we compare the peak zT value of KMgSb and KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 in the p-type region, an increase of ∼ 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='14% and 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='14% is achieved without and with SOC after alloy engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Although the SOC affects the zT value adversely, alloy engineering seems to be a promising route to induce neg- ative pressure and thus improve the zT dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' One important aspect of thermoelectric transport is the electronic relaxation time, which is a complex phe- nomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' As discussed above, the CRTA was considered to be the most convenient and helpful choice in electronic transport calculations, but it comes with a cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Under CRTA, the Seebeck and Hall coefficients become inde- pendent of scattering rates [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Moreover, a constant value of τ cannot account for all the scattering processes involved in electronic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' To get an insight, we 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) ne = 1019 cm 3 T = 300 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) ne = 1019 cm 3 T = 600 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) ne = 1020 cm 3 T = 300 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) ne = 1020 cm 3 T = 600 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) nh = 1019 cm 3 T = 300 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) nh = 1019 cm 3 T = 600 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) nh = 1020 cm 3 T = 300 K ADP IMP POP total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 (eV) 10 18 10 16 10 14 10 12 10 10 (s) nh = 1020 cm 3 T = 600 K ADP IMP POP total 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The scattering mechanism dependent electronic relaxation time (τ) in KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5, calculated at 300 K and 600 K for 1019 cm−3 and 1020 cm−3 carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' have shown the scattering mechanism dependent relax- ation time (τ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The overall electronic relaxation time is found to lie between 10−14 s and 10−16 s, which arises primarily from the POP with a small contribution from ADP and IMP scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The average value of τ is found to be larger for n-type doping than for p-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The results discussed above explain the impact of pres- sure, SOC, and alloy engineering on thermal and elec- tronic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We observed a quantitative as well as a qualitative difference between the thermoelectric trans- port results obtained with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Moreover, the thermoelectric performance shows a huge improve- ment after Bi doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Our results suggest that the ther- moelectric figure of merit (zT) should not be labelled merely by a single number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Therefore, we propose that the maximum zT value of KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 at 600 K may be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='51 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='75, depending upon the SOC strength and carrier concentration obtained in an exper- imentally grown sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this work, we started with the investigation of the thermal and electronic transport properties of the KMgX family (X = P, As, and Sb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The large SOC strength and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='0 eV band gap prompted us to zero in on KMgSb for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We find that in KMgSb the thermoelectric transport is strongly dependent on the SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The peak zT value of KMgSb without SOC (at 600 K) is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='38 in the p-type region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' after the inclusion of SOC the peak shifts to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='24 in the n-type region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' It is encouraging to note that the κL of KMgSb de- creases under negative pressure, implying the applica- tion of negative pressure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' by doping with larger size atoms, would be helpful in improving the thermoelectric performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' This finding guided us to explore a new ma- terial by substitutionally doping Bi (in place of smaller size Sb) in KMgSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' The alloy engineering does improve thermoelectric performance dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' On 50% sub- stitutional doping of Bi in place of Sb atoms, the κL be- comes ultra-low and the zT without (with) SOC changes from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='39 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='14) to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='75 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='51) in the p-type region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' To conclude, KMgSb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='5 is a potential candidate for thermal barrier applications due to its ultra-low κL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Even though the SOC deteriorates the thermoelectric performance (in comparison to the results without SOC) and cannot be done away with in the materials contain- ing heavy elements, alloy engineering may come as a big respite as it can improve the thermoelectric performance dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work used the Supercomputing facility of IIT Roorkee established under National Supercomputing Mission (NSM), Government of India and supported by Centre for Development of Advanced Computing (CDAC), Pune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' We have also used other computational facilities provided by Institute Computer Center (ICC), IIT Roorkee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' VC wish to acknowledge the financial sup- port received from Ministry of Education, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 7 [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Snyder and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' S.' metadata={'source': 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+page_content=' He, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Ying, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhao, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Energy Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 5, 1500411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhao, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Chen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Ren, Nano Energy 7, 97 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' He, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Hogan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Uher, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhao, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 29, 1702676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' He, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Hogan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Uher, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Dravid, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Kanatzidis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 134, 7902 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' [11] Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 97, 283 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Chang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Tan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Kanatzidis, Energy Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Chen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Ren, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' Energy Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} +page_content=' 3, 1195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfPAw4/content/2301.04969v1.pdf'} 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Ranasinghe +The University of Adelaide +Australia +damith.ranasinghe@adelaide.edu.au +ABSTRACT +Emulation-based fuzzers enable testing binaries without source +code, and facilitate testing embedded applications where automated +execution on the target hardware architecture is difficult and slow. +The instrumentation techniques added to extract feedback and +guide input mutations towards generating effective test cases is at +the core of modern fuzzers. But, modern emulation-based fuzzers +have evolved by re-purposing general-purpose emulators; conse- +quently, developing and integrating fuzzing techniques, such as +instrumentation methods, are difficult and often added in an ad-hoc +manner, specific to an instruction set architecture (ISA). This limits +state-of-the-art fuzzing techniques to few ISAs such as x86/x86-64 +or ARM/AArch64; a significant problem for firmware fuzzing of +diverse ISAs. +This study presents our efforts to re-think emulation for fuzzing. +We design and implement a fuzzing-specific, multi-architecture +emulation framework—Icicle. We demonstrate the capability to +add instrumentation once, in an architecture agnostic manner, with +low execution overhead. We employ Icicle as the emulator for a +state-of-the-art ARM firmware fuzzer—Fuzzware—and replicate +results. Significantly, we demonstrate the availability of new in- +strumentation in Icicle enabled the discovery of new bugs. We +demonstrate the fidelity of Icicle and efficacy of architecture ag- +nostic instrumentation by discovering LAVA-M benchmark bugs, +requiring a known and specific operational capability of instrumen- +tation techniques, across a diverse set of instruction set architectures +(x86, ARM/AArch64, RISC-V, MIPS). Further, to demonstrate the +effectiveness of Icicle to discover bugs in a currently unsupported +architecture in emulation-based fuzzers, we perform a fuzzing cam- +paign with real-world firmware binaries using Texas Instruments’ +MSP430 RISC ISA and discovered 7 new bugs. +CCS CONCEPTS +• Software and its engineering → Software testing and debug- +ging; • Security and privacy → Embedded systems security. +KEYWORDS +Fuzzing, emulation, embedded systems +ACM Reference Format: +Michael Chesser, Surya Nepal, and Damith C. Ranasinghe. 2023. Icicle: +A Re-Designed Emulator for Grey-Box Firmware Fuzzing. In Proceedings +of ACM SIGSOFT International Symposium on Software Testing and Anal- +ysis (ISSTA 2023). ACM, New York, NY, USA, 12 pages. https://doi.org/ +XXXXXXX.XXXXXXX +1 +INTRODUCTION +Fuzzing is an automated software testing methodology that re- +peatedly executes a program with generated inputs and monitors +execution for adverse behaviors. Progress in the field has greatly en- +hanced the bug discovery capability of modern fuzzers and fuzzing +is now widely used in the software development industry. In partic- +ular, grey-box fuzzing (or feedback-driven) methods have proven +to be highly effective at scale and are capable of finding bugs in a +diverse set of software [3, 4, 7, 31, 38, 44, 50, 53]. Grey-box fuzzing +relies on the ability to add instrumentation to the target binary +to obtain feedback. This feedback allows input generation to be +intelligently guided, improving a fuzzer’s ability to discover bugs. +A simple method to facilitate grey-box fuzzing is for the compiler +to inject instrumentation into the source code during compilation. +However, it is often necessary to fuzz binaries where source code +is unavailable—binary-only fuzzing—or where the target hardware +is not suitable for automating testing and testing is carried out +on a host machine with a different instruction set architecture +(ISA)—cross-architecture fuzzing. For instance, it is extremely chal- +lenging to perform rapid execution on devices typically used for +Internet of Things (IoT) applications and embedded systems in gen- +eral [11, 14, 22, 29, 30, 34, 39, 45, 66]. Consequently, we are forced +to use emulators capable of executing binaries built for the target on +a more convenient host machine; exploiting the resource capabili- +ties of the host for software bug discovery and triaging. Therefore, +emulators play a critical role in supporting both binary-only and +cross-architecture fuzzing. Significantly, emulators enable unpar- +alleled control and introspection over program execution, even +without source code and access to the original hardware. +Current state-of-practice for emulation-based grey-box fuzzing, +driven by its more recent evolution compared to emulators, is to +integrate fuzzing instrumentation into existing general-purpose em- +ulators. But, this can be challenging because these emulators were +not designed to support such modifications [51]. Consequently, +existing emulation-based fuzzers implement instrumentation tech- +niques either manually, through direct modifications to the emu- +lator [24, 25, 51, 64], or through limited interfaces that are unable +to support more advanced instrumentation [29, 39, 40, 49]. Further, +the absence of a consistent approach to add new, experimental, +instrumentation and the need for domain expertise in emulator de- +velopment to evaluate new fuzzing techniques are arguably barriers +to developments in the field. As a result, the benefits of extensive +research efforts to develop state-of-the-art fuzzing techniques can +remain limited to a specific ISA; this is undesirable. +Our Contributions. This study presents our efforts to design and +build a new multi-architecture emulation framework explicitly for +fuzzing. +In summary, we make the following contributions: +• We designed a new multi-architecture emulator, Icicle, for +directly supporting emulation-based fuzzing: i) designing for +arXiv:2301.13346v1 [cs.CR] 31 Jan 2023 + +ISSTA 2023, 17-21 July, 2023, Seattle, USA +Michael Chesser, Surya Nepal, and Damith C. Ranasinghe +architecture-agnostic instrumentation; ii) employing a decou- +pled design, enabling emulation, instrumentation and instruc- +tion set architecture (ISA) support to be maintained separately; +and iii) byte level memory-management to better support em- +ulating memory in embedded systems. +• We implemented Icicle as a coverage-guided greybox fuzzer +by integrating with AFL++ and Fuzzware. +• We conducted extensive experiments across five diverse ISAs, +multiple instrumentation techniques, 21 real-world binaries +and a synthetic test program. Notably, we demonstrate: i) the +instrumentation requirements of state-of-the-art fuzzing tech- +niques are satisfied with a unified instrumentation interface +without the need for architecture-specific knowledge; ii) the +effectiveness of Icicle by comparing against existing emula- +tors for the challenging task of firmware fuzzing. Significantly, +Icicle, supported by additional architecture-agnostic instru- +mentation, uncovered seven previously undiscovered bugs; +and iii) the efficacy of Icicle and its architecture agnostic in- +strumentation on a new ISA—we fuzz and discover new bugs +in real-world binaries written for Texas Instruments’ MSP430 +RISC architecture. Importantly, this ISA is currently not sup- +ported by existing emulation-based fuzzers. +• We open source1 Icicle to facilitate further improvements +and advance the field of emulation-based fuzzing in general. +2 +INSTRUMENTATION CHALLENGES IN +EMULATION-BASED FUZZING +In this section, we present an overview of different instrumentation +techniques that are used in modern grey-box fuzzing frameworks. +Subsequently, we highlight the issues hindering their implemen- +tation in existing general-purpose emulators without resorting to +direct, architecture-specific, modifications of the emulator that mo- +tivate the need of a re-designed emulator for fuzzing. +2.1 +Instrumentation Techniques +Grey-box fuzzers rely on instrumentation techniques to obtain feed- +back to enable more effective exploration of a target program which +is necessary for uncovering deeper bugs. Therefore, it is important +for fuzzing frameworks to support a diverse set of instrumentation +techniques. In this section we describe a number of instrumentation +techniques developed in previous research efforts. +Code coverage (Branch hit counts). Almost all grey-box fuzzers +utilize a form of code coverage for feedback. Code coverage identi- +fies inputs that reach new locations within a program by instrument- +ing the target so it notifies the fuzzer when new code is reached. +It is a proven and effective method that enables fuzzer to incre- +mentally discover different parts of the program [8, 44]. Branch hit +counts is an approach to code coverage popularized by AFL [64]. +This approach maintains a global map of 8-bit counters that are in- +cremented whenever an edge in the program is hit. After execution, +the values of each of the counters grouped into one of 8 ranges and +if any of the counters contain a value with a unique range, the fuzz +input corresponding to the execution is considered novel. +1https://github.com/icicle-emu/icicle +Context-sensitive branch coverage [12, 62]. Context-sensitive +branch coverage augments branch hit counts by hashing the edge +index with the current calling context. This allows the fuzzer to +obtain better feedback from branches inside of frequently called +functions. +CmpLog [25]. For many target binaries, code coverage is insuffi- +cient at finding inputs that reach deep parts of the program. For +example, comparisons against large constants (such as Listing 1) +are difficult to satisfy with just code coverage, because there is no +feedback mechanism that allows for incremental progress towards +solving the comparisons. +if (tag == 0x31677562) { +crash(); +} +Listing 1: Example program where crash is hard to reach +with traditional code coverage instrumentation. +One approach explored in several recent studies [5, 23, 28] is +to directly instrument the operands of comparisons. CmpLog is +a comparison tracing technique implemented in AFL++ based on +Redqeen [5] and Weizz [23]. CmpLog identifies comparisons +within a program, then adds instrumentation that captures the +operands of the comparison. After execution, the fuzzing frontend +can then scan the input for the operands in order to replace them +with their correct value (a process referred to as input-to-state +replacement). +CompareCov [25]. Is an alternative approach to solving complex +comparisons based on the existing compiler instrumentation tech- +nique, LAF-Intel [33]. CompareCov provides better feedback by +instrumenting the program to split comparisons involving large +values into comparison between individual bytes. The split com- +parison can subsequently be solved with code coverage instrumen- +tation. In addition to instruction-level comparisons, CompareCov +also attempts to improve feedback for memory comparison func- +tions (memcmp, strcmp and strncmp) by adding instrumentation to +update the coverage (bitmap) for every matching byte within the +comparison operation. +2.2 +Instrumentation Challenges +QEMU and Unicorn (forked from QEMU), have become the de facto +emulators for fuzzers [13, 14, 16, 22, 29, 30, 39, 40, 45, 51, 52, 58, 59, +65–67]. Therefore, for the remainder of this paper we will primarily +compare Icicle against emulator-level capabilities in QEMU and +Unicorn to support fuzzing. +QEMU was primarily designed for fast, general-purpose, emula- +tion, not for fuzzing. As a consequence, many design decisions differ +from those important for fuzzing. Crucially, it is difficult to add +advanced instrumentation techniques in an architecture agnostic +manner, for the following key reasons: +• QEMU’s intermediate representation for dynamic translation, +Tiny Code Generator (TCG) ops, is not designed for direct +analysis or manipulation [51]. Consequently, it is challenging +to add instrumentation without making invasive code changes +at a very low level. + +Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing +ISSTA 2023, 17-21 July, 2023, Seattle, USA +• QEMU is monolithic in design, providing little support for +extensibility, and assumes that it controls the full life cycle of +the emulation process. These properties inhibit the maintain- +ability of modifications and reduce scalability as it makes it +challenging to share state across fuzzing instances. +• Given QEMU’s historical focus on emulation, there is no uni- +fied mechanism for adding instrumentation. For example, code +coverage is implemented by modifying the code-generator to +inject code at the start of every basic block; more complex +techniques, such as CmpLog, modify the translation process of +individual architectures. Hence, adding new instrumentation +techniques require extensive emulation domain expertise. +Unicorn is a fork of QEMU designed for flexible and modular +CPU emulation. Unicorn extracts the CPU emulation component +from QEMU, configures it to always use the software MMU im- +plementation, and introduces hooks, functions that are called in +response to selected emulator events, such as memory accesses and +breakpoints [39, 40, 52]. +Unicorn’s function hooking API, enables fuzzers to inject func- +tions calls to observe the CPU state, which can be used to implement +some instrumentation techniques. However, the observable state +is architecture specific and Unicorn provides no support for ana- +lyzing the code semantics, making more advanced instrumentation +difficult. +The maintainers of the Unicorn project have also reported that it +is difficult to keep Unicorn up-to-date with improvements because +of QEMU’s monolithic design [48]. +Therefore, we are motivated to build a new multi-architecture +emulation framework explicitly designed for fuzzing; with the +ability to support sophisticated instrumentation methods in an +architecture-agnostic manner to enable fast emulation-based fuzzing +of binaries. +3 +ICICLE DESIGN AND IMPLEMENTATION +We provide an overview of our fuzzing specific multi-architecture +emulation framework in Figure 1. To enable architecture agonostic +fuzzing we use a portable intermediate representation (IR) that +is suitable for both emulation and program analysis. Translation +of the guest ISA to the portable IR is achieved using processor +specifications that are external to the emulator. This ensures that +architecture-specific details are kept decoupled; enabling fixes for +specification bugs and the addition of new architectures to be im- +plemented with minimal changes to the core emulator. Further, +in contrast to existing emulation-based fuzzing frameworks, we +define new instrumentation application programming interfaces +(APIs) that enable instrumentation to exist entirely outside of the +emulator. This facilitates researchers both in developing new instru- +mentation techniques for emulation-based fuzzers without domain +expertise in emulator development, and the immediate availability +(SLEIGH +specifications) +x86-64 +ARM +MIPS +1 +JIT invalidation +ICICLE +Instruction +Optimizer +Decoder +Lifter +Guest ISA +Instructions +P-code +Block translator +JIT cache +Snapshot +Store +JIT function +Block map +P-code block +Modified +blocks +New block +event +JIT Compiler +Code generation +(Cranelift) +Software MMU +P-code +Emulator +Environment Emulator +(Linux/MSP430 MCU) +Loader +(.elf/.bin/.ihex) +Virtual +Registers +Dispatcher +TLB +Tracer +Plugins +3 +4 +5 +6 +7 +8 +Code +coverage +CmpLog +CompareCov +2 +SLEIGH runtime +Figure 1: Overview of the core components in our fuzzing specific multi-architecture emulation framework. The instrumen- +tation workflow is as follows: 1 On initialization the emulator loads the appropriate SLEIGH processor specification, config- +uring the SLEIGH runtime. 2 One or more Tracer Plugins are registered with the emulator to support the instrumentation +needs of the fuzzer. 3 Once configured, the emulator loads the target binary into memory and starts execution. 4 During +execution, whenever the emulator attempts to execute a new instruction, the dispatcher initiates the translation process. 5 +Each guest instruction is then translated to P-code. 6 P-code operations are grouped into a block, and the block is stored in +a global block map. 7 Tracer Plugins are notified to allow them to analyze the new block and modify any block in the block +map required for instrumentation. 8 Any new or modified blocks are then compiled to native code using the JIT compiler +and the emulator resumes execution. Notably we implemented the components in the grey area and Icicle is more than 32,000 +lines of code. + +ISSTA 2023, 17-21 July, 2023, Seattle, USA +Michael Chesser, Surya Nepal, and Damith C. Ranasinghe +of these techniques across ISAs in a design-build-and-test once-only +paradigm. +3.1 +Fuzzing Specific Emulator Design +Icicle supports fast, multi-architecture, CPU emulation through +portable dynamic translation. First, guest ISA instructions are trans- +lated to an intermediate representation (IR) called P-code. P-code is +then just-in-time (JIT) compiled to the host architecture allowing +for efficient execution. +Icicle performs translation to P-code through the use of a SLEIGH2 +processor specification for the guest ISA. SLEIGH is a domain- +specific language (DSL) that describes how to decode and translate +the semantics of machine code into P-code. We chose SLEIGH as +the basis of Icicle’s CPU emulation for the following reasons: +• Broad architecture support. We leverage the diverse set of +SLEIGH specifications that have already been created as part +of the open-source Ghidra framework [60] (over 45 processor +kinds are supported). This enables Icicle to emulate a wide +range of architectures, including architectures unsupported by +other emulation frameworks like QEMU, with significantly +reduced effort. +• Designed for analysis. Unlike IRs used in other emulator frame- +works, P-code was explicitly designed to support program +analysis. For example, P-code maintains hints for call and +return operations even though such hints are unnecessary +for emulation. This makes it better suited for performing the +code analysis tasks required for advanced instrumentation +techniques. +• Suitable for emulation. P-code consists of small set of oper- +ations that can be efficiently executed by the host ISA. For +example, P-code avoids the use of bit-vectors. This allows for +fast emulation. +• Decoupling and ease of maintenance. Icicle is able to use the +original SLEIGH specifications written for Ghidra without any +modifications. This allows any improvements or fixes made +to a specification to be immediately usable by Icicle without +any changes to the core emulator. +Icicle implements a custom P-code emulator3 consisting of sev- +eral components, a custom SLEIGH runtime, a JIT-based execution +engine and a software memory-management unit (MMU). The +SLEIGH runtime handles loading the appropriate SLEIGH specifi- +cation for the guest architecture, assigning a mapping from guest +registers to virtual P-code registers, and then decoding and translat- +ing ISA-specific machine code to P-code. Unlike Ghidra’s SLEIGH +runtime, Icicle’s runtime assigns sequential IDs to virtual regis- +ters, allowing them to be managed in a dense array, improving +performance. We also implement a lightweight P-code optimization +pass that performs constant evaluation and dead-code elimination, +significantly reducing the amount of P-code operations when val- +ues are known at translation time. Icicle’s JIT-based execution +engine, then groups P-code operations in blocks and compiles them +to native code using Cranelift [1], an open-source low-level code +2The name SLEIGH was originally derived from SLED (Specification Language for +Encoding and Decoding), which also influenced the name of our emulator: Icicle +3Ghidra contains a limited P-code emulator and has been used for micro-fuzzing [19], +but is unable to satisfy the needs of a full modern fuzzing framework. For instance, +Ghidra’s P-code emulator is interpreter-based hindering performance. +generation framework. Cranelift provides register allocation, in- +struction legalization, and additional optimizations. Later, during a +recompilation step, multiple blocks are compiled as part of a single +compilation unit, enabling additional optimizations that improve +performance. Notably, unlike existing emulators, Icicle does not +discard the P-code representation of each block after JIT compi- +lation. This can significantly aid any analysis used for complex +instrumentation, at the cost of some additional memory overhead. +When the memory layout of the guest is incompatible with the +host, it is necessary for the emulator to handle the differences. +Therefore, Icicle uses a software MMU to handle guest memory +accesses (Figure 2). The software MMU maintains virtual mapping +table that allows guest memory and memory mapped IO (MMIO) +to be mapped in the emulator. The mapping table represented as +a range-map (implemented with B-Tree), allowing for byte-level +precision at the cost of more expensive lookups. To improve effi- +ciency, we cache translated addresses in a lookup table referred to +as a translation lookaside buffer (TLB), named after its hardware +analog. The JIT compiled code, can directly access guest memory +using the TLB allowing for fast execution in most cases. Whenever, +an address not in the TLB, is encountered, the JIT calls a runtime +helper that handles the access and caches the translated address. To +retain byte-level mapping, Icicle maintains a permission byte for +each physical byte which is checked by the JIT on access, similar +to approaches used in prior work [21, 24, 56]. Both Unicorn and +QEMU (when running in full-system mode) also implement a soft- +ware MMU. However, they both require memory to be mapped in +page-sized (4 KB) regions. +The added flexibility of the byte-level mapping in Icicle allows +more accurate emulation of embedded system memory, and can be +used to enable better bug detection. +Virtual mapping +TLB (direct mapped) +Physical +memory pages +Data 0 +Perm 0 +Data 1 +Perm 1 +... +Tag Pointer +Real memory +Mapped memory +(unallocated) +I/O handler +(unmapped) +I/O Callbacks +JIT access +Runtime +access +Figure 2: Overview of the byte-level software memory- +management unit (MMU) implemented in Icicle. +In addition to CPU emulation, most binaries interact with exter- +nal resources such as file systems, hardware, and other software. +Icicle is designed to be flexible and extensible through the use of +pluggable environment emulators (see Figure 1). To demonstrate the +functionality of the system, we have implemented environments to +allow comparisons against existing emulation-based fuzzers. The +current implementation supports fuzzing Linux userspace binaries + +Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing +ISSTA 2023, 17-21 July, 2023, Seattle, USA +by emulating a subset of system calls, supports fuzzing several +MSP430 MCUs ISAs, and supports embedded ARM binaries using +Fuzzware [52]. +3.2 +Architecture-Agnostic Instrumentation +To implement arbitrarily complex fuzzing instrumentation requires: +i) the ability to analyse the semantics of a target program; ii) an effi- +cient mechanism to capture runtime information about the running +program; iii) a way of sharing the captured information with the +fuzzing frontend. Additionally, to be effective in a fuzzing context, +these requirements must be supported in a manner that has low +performance overheads. +Icicle supports these requirements through a set of APIs added +to the emulator, we refer to instrumentation utilizing these APIs as +Tracer Plugins. These APIs enable: +• Direct access to the architecture-agnostic P-code representation +of the program. Plugins are able register a callback function +to be called whenever the emulator translates a new block to +P-code. The callback function is provided with the full code- +cache including the newly translated block, satisfying the first +requirement enabling architecture agnostic code analysis. +• Inline code-injection. Plugins can inject additional P-code op- +erations into any block enabling inline instrumentation to be +supported in an architecture agonstic manner. Any modified +blocks are invalided by Icicle and re-compiled by the JIT the +next time they are executed. +• Registry of JIT and fuzzer accessible shared memory. During ini- +tialization, plugins are able to register storage locations with +the emulator, which can later be manipulated with P-code op- +erations. Additionally, Icicle allows plugins to define custom +P-code registers, these custom registers are treated the same +as guest registers for the purpose of register allocation during +JIT code-generation, which can allow for more efficient instru- +mentation in some cases. This enables data to be efficiently +saved by injected instrumentation and analyzed as part of the +fuzzing loop. +To illustrate expressiveness Icicle’s instrumentation method +and it ability to support architecture-agnostic instrumentation, we +discuss the implementation of the four techniques discussed in +Section. 2.1 in Icicle and compare them to implementations in +other emulation-based fuzzers. +Branch hit counts. In Icicle, branch hit counts are implemented +by a Tracer Plugin that does the following: during initialization, it +registers the location of coverage bitmap with the emulator and de- +fines a custom register to store the previous program location. When +a new block is translated, the plugin injects code at the start of the +block that computes a hash of (current_location, previous_location), +which is then used as an index for updating the coverage bitmap. +Since the instrumentation is implemented using P-code injections, +the JIT is able to generate native code that updates the coverage +bitmap without resorting to a function call. In addition to branch +hit counts, Icicle also implements both block-only coverage and +edge coverage. +Existing emulators are also able to add branch hit count instru- +mentation in an architecture independent manner by injecting code +when new translation blocks are created, which is common across +architectures in QEMU. However, AFL++’s implementation in both +QEMU and Unicorn makes direct modifications the emulator (al- +though the changes are relatively minor). Notably, in AFL++’s Uni- +corn mode, branch coverage instrumentation is not implemented +using Unicorn’s hooking API, since the instrumentation is highly +performance sensitive and the hooking API imposes additional +overheads. +Context-sensitive branch coverage. Icicle implements context- +sensitive branch coverage with a Tracer Plugin. This plugin defines +a custom register to store the context, then when a new block ending +with a CALL is translated, the generates a random value to use as +context for the current location and XORs it the context register. In +the block after the call, the instrumentation is injected to clear the +added context. The branch hit count plugin is then modified to use +the context value by using it as part of computing the index into the +coverage map. Since the CALL hint is part of P-code representation, +it allows us to write a portable implementation that works across +architectures. +Context-sensitive coverage was first implemented in Angora[12] +using compiler-based instrumentation, and in afl-sensitive [62] for +binary-only instrumentation using QEMU. afl-sensitive’s implemen- +tation modifies QEMU’s x86 translation layer to add instrumenta- +tion that updates the calling context on call and ret instructions. +Since afl-sensitive instruments x86 specific instructions it is not +portable to other architectures. +CmpLog. There are two parts to CmpLog, first relevant comparison +operations must be identified, and second, the operands of each +comparison must be copied to a fuzzer accessible location. +Inspired by the success of Datalog for program analysis tasks [26, +57], we implement a comparison finding algorithm as a set of Data- +log rules in Listing 2. Since the rules are defined in terms of P-code +operations, it allows Icicle support CmpLog for any architecture. +% x is an copy of the destination of an operation. +copy(x, x) :- op(x, _, _, _). +% b = a if it is the destination of a copy-like op with a. +copy(a, b) :- op(b, "COPY", a). +copy(a, b) :- op(b, "ZXT", a). +% b = a if x = a and b = x +copy(a, b) :- copy(a, x), copy(x, b). +% Identify p-code operations corresponding to comparisons. +cmp("==", cond, a, b) :- op(cond, "==", a, b). +cmp("!=", cond, a, b) :- op(cond, "!=", a, b). +% `(a - b) [cmp] 0` => `a [cmp] b` (subtract and compare with zero) +cmp(op, cond, a, b) :- op(cond, "-", a, b), cmp(op, cond, x, 0). +% `!(a [inv(op)] b)` => `a [op] b` (inverted comparison) +cmp("!=", cond, x, y) :- op(notc, "!", cond), cmp("==", notc, x, y). +cmp("==", cond, x, y) :- op(notc, "!", cond), cmp("!=", notc, x, y). +% Output comparisons that flow into the branch condition. +output(op, a, b) :- cmp(op, cond, a, b), copy(cond, x), branch(x). +Listing 2: Datalog rules for finding comparison operands. +The list of p-code operations to analyse, and the branch exit +condition are provided as inputs. +In contrast, existing CmpLog implementations require identi- +fying architecture specific instructions in order to identify com- +parisons. For example, on x86, AFL++’s instruments CMP and SUB +instructions, by modifying QEMU’s translation stage. This has two +main issues: 1) since the instrumentation looks for specific instruc- +tions, a separate implementation is required for each architecture, + +ISSTA 2023, 17-21 July, 2023, Seattle, USA +Michael Chesser, Surya Nepal, and Damith C. Ranasinghe +2) it can result in excessive instrumentation, for example most SUB +operations on x86 are not used for comparisons. CmpLog is not +supported in Unicorn. +CompareCov. In Icicle, integer comparisons are identified using +the same algorithm as CmpLog. Once identified, Icicle injects code +that writes to the coverage bitmap for each matching byte before the +original comparison operation. For memory comparisons functions, +Icicle searches for the target functions in the program’s symbol +table and injects instrumentation when a block calling the target +function is translated. This allows Icicle’s instrumentation to be +used for statically linked binaries including firmware (as long as +the symbol table has not been stripped). +In contrast, AFL++’s implementation for integer comparisons +requires identifying architecture specific comparison instructions, +like CmpLog instrumentation. Further, for memory comparison +functions, it relies on the dynamic linker to replace the original +comparison functions with instrumented versions. This approach is +unable to support instrumenting statically linked firmware binaries. +Summary. Each instrumentation technique is implemented +targeting P-code enabling it to support any ISA. And, as an +added benefit, only knowledge of P-code is adequate for devel- +oping new instrumentation techniques. +3.3 +Fuzzing Frontend Integration +Modern grey-box fuzzing frameworks consists of two main compo- +nents: the frontend which handles input generation, input sched- +uling, hang detection and crash deduplication, and the backend +which manages program execution, crash monitoring, and instru- +mentation. Emulation-based fuzzers utilize emulators as the fuzzing +backend allowing for binary-only and cross-architecture fuzzing. Ici- +cle is a new fuzzing backend, therefore, we make our emulator +compatible with an existing fuzzing framework: AFL++ [25] to +avoid implementing a new frontend. AFL++ is a state-of-the-art +fuzzing framework derived from the well-known American Fuzzy +Lop (AFL) [64] project, with general improvements, and support for +additional fuzzing techniques. Icicle integrates with AFL++ using +the forksever interface also used by AFL++’s QEMU-mode. +4 +EVALUATION +Settings. Unless otherwise specified, all experiments were carried +out with AFL++ 4.01a as the fuzzing frontend on an AMD Ryzen +Threadripper 3990X restricted to a single core. All AFL++ settings +were kept as default, except to enable instrumentation as needed +and to adjust the timeout for hang detection. +Experiments. We design our experimental regime to answer five +specific questions articulated in Section 4.1-4.5. +4.1 +Is Icicle’s instrumentation portable across +diverse ISAs? +To ensure that architecture-agnostic instrumentation implemented +in our emulator is operational across a range of architectures, we +designed a test program, shown in Listing 3, that consists of 5 +synthetic bugs designed to test specific instrumentation. +void test_instrumentation(char* buf) { +// (1) comparison against a single byte in the input +if (buf[0] == '%') { +crash(1); +} +// (2) Multiple comparison against single bytes of the input. +if (buf[0] == 'i' && buf[1] == 'x' && +buf[2] == 'S' && buf[3] == 'D') { +crash(2); +} +// (3) A single comparison against multiple input bytes. +if (*(u32*)buf == *(u32*)"wzfc") { +crash(3); +} +// (4) A multi-byte comparison across a function call. +if (compare(buf, "dGlIHF1W") == 0) { +crash(4); +} +// (5) Saturate coverage then compare. +saturate_compare2_cov(); +u32 tmp = *((u32*)buf) ^ 0x46092d5f; +if (compare2(tmp, 0x7451496b)) { +crash(5); +} +} +Listing 3: Test program used for evaluating instrumenta- +tions. +We evaluate the portability of Icicle’s instrumentation by fuzzing +the test program compiled for 5 different architectures. For archi- +tectures with Linux support, we configure the program to read the +input from stdin. For MSP430, the program reads from a peripheral +mapped to the fuzzing input. After compiling the binary for each +architecture, we manually verified that the machine code of out- +put binary behaves as expected. As a baseline we compare against +AFL++’s QEMU-mode when instrumentation is supported for the +guest architecture. For each fuzzing configuration, we perform 20 +trials for a maximum of 10 minutes starting with an uninformed +seed. The results from this experiment are shown in Table 1. +Both Bug1 and Bug2 are discoverable with code coverage alone, +so are found by all fuzzing configurations. Bug3 requires additional +instrumentation to be found so can only be found when one of the +two comparison instrumentation techniques is enabled, except for +the MSP430 binary. With CmpLog, the fuzzer can find a solution +via an input-to-state mutation directly replacing the incorrect value +with a correct one, with CompareCov enabled, the comparison is +“split” into byte-level comparisons, and the fuzzer observes incre- +mental coverage feedback similar to Bug2. Since MSP430 is a 16-bit +architecture, the compiler splits the 32-bit comparison into two 16- +bit comparisons allowing the fuzzer to eventually find the crashing +input for Bug3 without additional instrumentation. Bug4 evaluates +the fuzzers ability to solve memory comparison functions so is only +discovered when CmpLog instrumentation enabled, which gener- +ally finds the crashing input within seconds. CompareCov fails to +find the bug, since compare is not a standard comparison function +and is therefore not instrumented by CompareCov. CmpLog is only +partially implemented for QEMU on AArch64 (function calls are not +instrumented) so fails to find Bug4. Bug5 tests the fuzzer’s ability to +find a bug in a function where code coverage is saturated by a pre- +vious call, so is only found when context sensitive branch coverage +is enabled, which only Icicle supports on all architectures. + +Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing +ISSTA 2023, 17-21 July, 2023, Seattle, USA +Table 1: Results from different fuzzing instrumentation configurations for the test program. ✓ denotes the bug ID was found +at least once within 10 minutes. Each test was repeated 20 times. Shaded grey areas are due to: i) unsupported fuzzing instru- +mentation for MIPS and RISC-V in QEMU emulation with AFL++; and ii) MSP430 ISA being unsupported in QEMU. +x86-64 +AArch64 +MIPS +RISC-V +MSP430 +Fuzzer +Instrumentation +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +Cov +✓ +✓ +- +- +- +✓ +✓ +- +- +- +✓ +✓ +- +- +- +✓ +✓ +- +- +- +✓ +✓ +✓ +- +- +Cov+CmpLog +✓ +✓ +✓ +✓ +- +✓ +✓ +✓ +✓ +- +✓ +✓ +✓ +✓ +- +✓ +✓ +✓ +✓ +- +✓ +✓ +✓ +✓ +- +Cov+CompareCov +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +Icicle (ours) +Cov+Context +✓ +✓ +- +- +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +✓ +✓ +✓ +✓ +- +✓ +Cov +✓ +✓ +- +- +- +✓ +✓ +- +- +- +✓ +✓ +- +- +- +✓ +✓ +- +- +- +Cov+CmpLog +✓ +✓ +✓ +✓ +- +✓ +✓ +✓ +- +- +QEMU +Cov+CompareCov +✓ +✓ +✓ +- +- +✓ +✓ +✓ +- +- +Summary The test program binaries for five different ISAs +provide empirical evidence that the architecture agnostic in- +strumentation implementation of the different instrumenta- +tion techniques in Icicle is both effective and portable across +architectures. +4.2 +Is architecture-agnostic instrumentation as +effective as existing architecture-specific +implementations? +LAVA-M [20] is a widely used set of binaries for evaluating and +benchmarking fuzzers. It consists of four binaries from GNU core- +utils [27] each injected with synthetic bugs. While the injected bugs +are not representative of typical real-world vulnerabilities [35], pre- +vious work has demonstrated that these bugs are difficult to find +with code coverage only, however can be found with by instru- +menting comparison operations [5, 12, 28]. This naturally lends +itself to assessing Icicle’s architecture-agnostic implementation of +CmpLog and CompareCov instrumentation. +Using AFL++ as the frontend, we evaluate the bug discovery +capability of Icicle across four different ISAs (x86, AArch64, MIPS, +and RISC-V), the first two of which we compare against QEMU4. +We also evaluate both QEMU and Icicle on x86 with code coverage +only as a baseline. For each of the injected bugs, a unique ID is +written to stdout whenever the bug is triggered. Therefore, we +can verify each crash by running the x86 version of the binary on +the host machine then checking for unique bug IDs in stdout. We +perform 5 trials for each fuzzing configuration running for 12 hours +each, starting with the same two initial seeds as Section 4.1. The +results of the benchmark are shown in Figure 3. +With code coverage alone almost no bugs are found by either +emulator in any of the binaries. Both comparison instrumentation +techniques allow most bugs to be found, with CmpLog finding bugs +significantly faster than CompareCov in several cases. Icicle’s re- +sults closely match QEMU results for both AArch64 and x86, which +supports our claim that Icicle’s instrumentation is as effective as +the architecture-specific approach employed by AFL++’s QEMU- +mode. On the two additional architectures tested with Icicle both +instrumentation techniques continue to be effective. However, the +4We compare with QEMU not Unicorn, since Unicorn cannot directly execute Linux +binaries. Additionally, since neither CmpLog nor CompareCov instrumentation are +supported in AFL++’s QEMU-mode for RISC-V and MIPS, we only evaluate these +architectures with Icicle +results for the MIPS version of uniq are slightly worse, this is +caused by differences in the memory layout (MIPS uses a 32-bit ad- +dress space, while the other architectures are 64-bit), which causes +issues when replaying the crashing input on the x86 host. +The differences in the number of crashes found for who binary +across architecture is caused caused by platform specific behaviour +in the program itself. The fuzz input is parsed as a utmpx structure, +however the layout of the fields within the structure is different +across architectures. This can cause certain bugs to become un- +reachable, and also causes issues when we attempt to replay the +crashing inputs on the x86 version of the binary in order to verify +the crash IDs. Additionally, the binary frequently crashes before +a bug ID is flushed to stdout (caused by internal buffering), which +prevents us from obtaining the bug ID from the original execution. +Notably, all bugs reported and discovered are those reproduced on +both the guest architecture and the host (x86). This is additional +evidence of the importance of binary-only and cross-architecture +fuzzing; even when source code is available, program behaviour +can differ on between architectures. +Summary Discovering LAVA-M benchmark bugs require a +specific operational capability from instrumentation tech- +niques to solve comparison operations; namely CompCov +or CompLog. Icicle’s results closely match QEMU results for +both AArch64 and x86, supporting our claim that Icicle’s +instrumentation is as effective as the architecture-specific +approach employed by AFL++’s QEMU-mode. On MIPS and +RISC-V architectures (where AFL++’s QEMU-mode does not +support the necessary instrumentation) both instrumentation +techniques tested with Icicle continue to be effective. +4.3 +Can Icicle be used to implement and +enhance state-of-the-art firmware fuzzing +techniques? +Fuzzware [52] is a recent state-of-the-art fuzzing framework for +analyzing ARM firmware binaries. Fuzzware extends Unicorn to in- +strument and execute ARM firmware. We replace Unicorn with Ici- +cle to evaluate Icicle’s ability to support state-of-the-art firmware +fuzzing. We then tested our modified version (Fuzzware-Icicle) by +attempting to reproduce Fuzzware’s results on the 10 binaries used +in the P2IM [22] firmware set as they were evaluated extensively + +ISSTA 2023, 17-21 July, 2023, Seattle, USA +Michael Chesser, Surya Nepal, and Damith C. Ranasinghe +by [22, 67] and Fuzzware. Importantly, since Icicle’s instrumenta- +tion is portable we are able to support additional instrumentation +when fuzzing ARM firmware. In particular, we perform additional +tests with CompareCov instrumentation enabled to allow for better +comparison solving5. We followed the same experimental setup for +Fuzzware as described in the original paper (we used the same +number of trials, seeds and run time duration). +We were able to successfully rediscover all 16 of the bugs found +by Fuzzware, and additionally, with CompareCov enable, Fuzzware- +Icicle was able to find an additional bug in the Console binary not +reported by any prior work6. As part of the rtc settime command, +the firmware reads a date from the user in the form YYYY-MM-DD +HH:MM:SS without checking whether the parsed date is valid. This +causes an out-of-bounds access when the name of the month is +resolved using a lookup table. Since reaching this bug requires first +solving a string comparison to reach the rtc handler, then solving a +second string comparison for the settime subcommand, we believe +the added instrumentation was critical to finding this bug. +Icicle also found an additional crash in the Soldering Iron bi- +nary. At high temperatures, rendering the heat indicator causes the +buffer allocated for the LCD screen to overflow. However, after fur- +ther analysis we discovered the maximum temperature is restricted +in software, indicating that the bug is a false-positive caused by +Fuzzware’s peripheral modelling strategy. +In addition to reproducing the bugs, we also investigated whether +Fuzzware-Icicle is able to maintain the same level of block cov- +erage as the original implementation. The results are shown in +Figure 4. For almost all of the evaluated binaries we achieve almost +identical block coverage to Fuzzware, with some small differences +5We did not test with CmpLog, since effective use of the instrumentation requires +additional integration with fuzzing frontend, unsupported by Fuzzware. +6Crashing inputs for each of the discovered bugs are available in our GitHub repository. +of which we manually investigated. With CompareCov enabled, +Fuzzware-Icicle achieves higher coverage in two of the binaries: +Console, and Steering Control. The higher coverage in Console +corresponds to reaching different command handlers that are dis- +patched based on string comparisons, including the sub-commands +of the rtc handler that contains the bug discussed above. Simi- +larly, Steering Control contains two commands, that are triggered +when the matching string is read by the firmware ("steer", and +"motor"). CompareCov enables Icicle to generate inputs contain- +ing the command strings, and thus is able to reach additional code. +The discrepancies in the Gateway and Soldering Iron binaries are +caused by high variance between fuzzing runs, running additional +trials would likely remove any discrepancies. +Summary Icicle is a robust emulator capable of supporting +the current state-of-the-art ARM firmware fuzzer, Fuzzware. +We discovered all 16 known bugs. Icicle provides a direct +substitute for Unicorn with the added advantage of additional, +architecture agnostic instrumentation shown to be effective +at improving coverage and discovering 2 new bugs not reported +by prior work. +4.4 +Can Icicle discover bugs in real world +binaries in an ISA currently not supported +by emulation-based fuzzers? +To demonstrate the architecture-independent benefits of our proto- +type emulator, we investigate fuzzing firmware written for MSP430 +microcontrollers. FiE [17], is the only prior study that attempted to +find bugs in MSP430 firmware. However, FiE requires C source-code +and therefore does not support manually written assembly code +(which is common in larger firmware), and is incapable binary-only +x86-64 +(Cov+CmpLog) +(Cov+CompCov) +(Cov+CmpLog) +AArch64 +(Cov+CompCov) +MIPS +(Cov+CmpLog) +(Cov+CompCov) +RISC-V +(Cov+CmpLog) +(Cov+CompCov) +md5sum +uniq +who +0 +3 +6 +9 +12 0 +3 +6 +9 +12 +0 +3 +6 +9 +12 0 +3 +6 +9 +12 +0 +3 +6 +9 +12 0 +3 +6 +9 +12 +0 +3 +6 +9 +12 0 +3 +6 +9 +12 +0 +20 +40 +0 +20 +40 +60 +0 +10 +20 +30 +0 +1000 +2000 +Duration (h) +Bugs found +ICICLE +QEMU +base64 +Figure 3: LAVA-M bugs found over time in each binary. The solid line represents the median number of bugs found, the shaded +area represents the min/max coverage across all trials, and the black dotted lines represent the number of bugs listed in the +LAVA-M paper (Note: it is well known that it is possible to trigger additional bugs other than specified in the original paper). + +Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing +ISSTA 2023, 17-21 July, 2023, Seattle, USA +PLC +Reflow Oven +Robot +Soldering Iron +Steering Control +CNC +Console +Drone +Gateway +Heat Press +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +4 +8 +12 +16 +20 +24 +0 +200 +400 +0 +200 +400 +600 +0 +1000 +2000 +3000 +0 +1000 +2000 +0 +500 +1000 +1500 +0 +500 +1000 +0 +300 +600 +900 +0 +250 +500 +750 +1000 +1250 +0 +1000 +2000 +0 +200 +400 +600 +Duration (h) +#Blocks +ICICLE +Unicorn +ICICLE (CompareCov) +Figure 4: Block coverage over time for ARM firmware using the two different emulators and CompareCov instrumentation +supported in Icicle. The solid line represents the median coverage of 5 runs, and the shaded area represents min/max coverage. +Table 2: Discovered vulnerabilities in MSP430 ISA binaries. +Firmware +Bug description (PoCs & stack traces on GitHub) +Goodwatch +Incorrect comparison when writing to log buffer. +Goodwatch +Buffer overflow when handling zero length packet. +Goodwatch +Stack overflow in RNG generation. +Goodwatch +Out-of-bounds access in OOK keypress. +Goodwatch +Out-of-bounds access in Stopwatch. +H4_PacketProtocol +Unchecked Interface Index in Get Descriptor. +H4_PacketProtocol +Buffer overflow in Set Report. +fuzzing. Further, MSP430 firmware is not supported by any exist- +ing emulation-based fuzzing framework7, and therefore presents a +compelling use case for fuzzing with Icicle. +Similar to existing monolithic firmware fuzzing approaches [22, +52], we handle peripheral accesses for MSP430 firmware by read- +ing them from the fuzzer input. We found this highly effective at +finding bugs. We selected 3 different firmware to evaluate. First, +inspired by FiE, we evaluated the USB SDK provided as part of +TI’s MSP430 USB Developers Package8 using example programs +provided as part of the development package (H4_PacketProtocol) +as a harness. Second, we compiled an unmodified version of the +Goodwatch [61] firmware9, a hardware and firmware replacement +for Casio calculator watches based on the CC430 MCU (a MSP430 +CPU with an integrated RF transceiver). Additionally, we investi- +gated Icicle’s ability to test closed-source firmware by extracting +the firmware off a commercial medical device, a Polar heart rate +tracker, containing a MSP430F2132 MCU. +The block coverage results are summarized Figure 3. After triag- +ing the results, we identified two unique bugs H4_PacketProtocol +and 5 unique bugs in the Goodwatch firmware and 3 additional +crashes related to debugging features. While no bugs were discov- +ered for the Polar heart rate tracker, the fuzzer reached almost all +7[36] is a fork of QEMU adding MSP430 support, however is outdated, not integrated +with any fuzzing framework and does not support MSP430 CPUX extensions. +8https://www.ti.com/tool/MSP430USBDEVPACK version 5.20.06.03 +9commit: c8859f845fccf56585a127059b1d1b825b381673 +Table 3: Block coverage (#BB) for MSP430 binaries with and +without CmpLog instrumentation enabled. Avg represents +the median coverage achieved after 24 hours in 5 trials. +Firmware +#BB +total +Instrumentation +#BB +min +#BB +avg +#BB +max +Goodwatch +3263 +Cov +2336 +2362 +2441 +Cov+CmpLog +2438 +2503 +2526 +H4 Packet +Protocol +925 +Cov +819 +821 +891 +Cov+CmpLog +813 +910 +914 +Heart Rate +Tracker +744 +Cov +679 +679 +716 +Cov+CmpLog +680 +717 +718 +blocks in the firmware. The bugs discovered by Icicle are summa- +rized in Table 2, and for each bug discovered we provide input files +and a detailed crash analysis in our GitHub repository10. +Summary MSP430 firmware fuzzing is not supported by ex- +isting emulation-based fuzzing frameworks. Case studies with +Icicle and its suite of architecture agnostic instrumentation +discovered seven undiscovered software bugs in two (USB +SDK–H4_PacketProtocol, and Goodwatch) of the three tested +MSP430 binaries. +4.5 +How does Icicle perform in fuzz testing? +In the development of Icicle, we made efforts to ensure that Icicle +has good performance in general. We compared fuzz test execution +speed of Icicle with Unicorn (emulator) employed by the sate-of- +the-art fuzzer, Fuzzware, on the P2IM dataset [22] and summarise +the results in Figure 5. +Summary Icicle has approximately the same performance +as Unicorn for fuzzing monolithic firmware binaries. +10https://github.com/icicle-emu/icicle/crash-analysis + +ISSTA 2023, 17-21 July, 2023, Seattle, USA +Michael Chesser, Surya Nepal, and Damith C. Ranasinghe +PLC +Reflow Oven +Robot +Soldering Iron +Steering Control +CNC +Console +Drone +Gateway +Heat Press +0 +30 +60 +90 +120 +0 +20 +40 +60 +80 +0 +100 +200 +300 +0 +100 +200 +0 +50 +100 +150 +0 +100 +200 +0 +25 +50 +75 +100 +0 +10 +20 +30 +0 +50 +100 +150 +200 +250 +0 +50 +100 +150 +Execs/second +ICICLE +Unicorn +Figure 5: Icicle and Unicorn performance comparison when +integrated with the state-of-the-art fuzzer, Fuzzware. +5 +DISCUSSION AND LIMITATIONS +Although we have taken the first steps to re-think and re-design an +emulation framework to directly support fuzzing requirements, and +instrumentation development and testing, the current implemen- +tation is not without limitations. The released emulator prototype +was primarily designed for CPU ISA emulation, similar to the goals +of the Unicorn project. As a result, Linux emulation is minimal, +and more complex hardware emulation required for full-system +emulation (e.g., page-table emulation) is not currently supported. +5.1 +Emulator Correctness +In emulation-based fuzzing, since the program not executed on the +original hardware, there is a risk that any crashes discovered could +be caused by emulation issues, not bugs in the target program. To +reduce the chance of emulation bugs in Icicle, first, we employ a +differential testing strategy, similar to other widely used approaches +for testing CPU emulators [2, 32, 41–43, 63]. Second, we manually +investigated any crashes discovered in benchmark evaluations and +ensure they are caused by program bugs. +5.2 +Performance +In the development of Icicle, while we made efforts to ensure that +Icicle has good performance in general, there are a number of addi- +tional optimizations possible. The current implementation of Icicle +has demonstrably similar performance to Unicorn (see Figure 5). Al- +though a direct performance comparison against QEMU is desirable, +it is more difficult. Because, Icicle implements a forkserver similar +to AFL++’s persistent mode, however we run AFL++’s QEMU-mode +without this feature since (currently) persistent mode requires a +significant amount of manual effort to set up for each binary (no- +tably, Icicle’s implementation is automatic for Linux binaries). This +results in Icicle performing significantly faster for small binaries. +Icicle always translates memory accesses in software (like Uni- +corn) while AFL++’s QEMU-mode can utilize hardware address +translation when running Linux user-space binaries on a Linux +host resulting in a significant speedup for QEMU for larger Linux +binaries. +6 +RELATED WORK +Improving emulation-based fuzzing. There has been some ef- +fort in improving QEMU and Unicorn for fuzzing, including, im- +proving runtime performance [6, 25], enabling support for full- +system emulation of Linux-based firmware [11, 15, 59, 65, 66], and +extending the emulator to support additional analysis such as taint +tracking and symbolic execution [51, 58]. +Binary-only fuzzing. Without access to source-code it is chal- +lenging to use fuzzing techniques that rely on instrumentation, +since the simplest approach using compiler-based code injection, is +not possible. Fuzzers that support targets without source-code are +known binary-only fuzzers. Emulation-based approaches are one +solution, however there are several other alternatives. +Virtualisation/hardware-assisted approaches (e.g., kAFL [54], +and NyX [53, 55]) use a variety of hardware features to implement +fuzzing instrumentation. Since they require additional hardware +support some instrumentation cannot be easily implemented, and +firmware fuzzing is not supported. Static rewriters (e.g., Retrowrite [18], +Datalog Disassembly [26], Zafl [46]) disassemble a binary, inject +instrumentation, then reassemble the binary. This can enable close +to compiler-level instrumentation performance, however the com- +plexity involved in the rewriting process often results in correctness +issues, typically firmware binaries are not well supported, and static +rewriting cannot be used cross-architecture fuzzing. While, dy- +namic instrumentation tools (e.g., DynamoRIO [9], PIN [37], CMU +BAP [10], Valgrind [47]), share significant similarities to emulation- +based approaches, they are more restrictive than full emulators and +are unable to support firmware fuzzing. +Embedded system fuzzing. Fuzzing embedded systems and IoT +devices is difficult because we cannot avoid dealing with hardware +and peripheral interactions since it might represent a majority of +the code we are trying to test. As a result, emulation-based fuzzers +need to support more than just CPU emulation. Past work has +extended either QEMU or Unicorn to support firmware fuzzing +through, hardware-in-the-loop approaches [16], peripheral model- +ing [17, 22, 30, 67], or emulating the hardware abstraction layer [14]. +More recently, Fuzzware [52], outperformed prior firmware fuzzing +approaches by automatically generating peripherals models using +local symbolic execution. Notably, aside from FiE (which is source- +based and only targets MSP430), all these approaches only evaluate +firmware written for the ARM architecture. Icicle makes it eas- +ier to fuzzing multiple architectures, which we hope will assist in +increasing the architecture diversity in future work. +7 +CONCLUSION +Emulation-based fuzzing techniques, supported by effective instru- +mentation, are highly flexible and are the only method for cross- +architecture fuzzing. For historical reasons, emulators used in ex- +isting emulation-based fuzzing frameworks were not designed for +fuzzing and has made it difficult to meet fuzzing specific needs such +as implementing advanced instrumentation techniques supporting +a design-build-and-test once-only paradigm across multiple ISAs, +and implementing fuzzing specific optimizations. + +Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing +ISSTA 2023, 17-21 July, 2023, Seattle, USA +We designed and implemented a new multi-architecture emu- +lation framework for fuzzing. Within our framework, we imple- +mented four different architecture agnostic instrumentation tech- +niques and demonstrated that a single architecture-independent +implementation is effective across multiple architectures. Our em- +ulation platform is extremely flexible, supporting a wide range +of ISAs, especially significant in fuzzing firmware in embedded +systems and IoT devices. 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Automatic Firmware +Emulation through Invalidity-guided Knowledge Inference.. In USENIX Security +Symposium. 2007–2024. + diff --git a/btFQT4oBgHgl3EQfhjY0/content/tmp_files/load_file.txt b/btFQT4oBgHgl3EQfhjY0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a407c5a237593592fd46db335464eba2425a060 --- /dev/null +++ b/btFQT4oBgHgl3EQfhjY0/content/tmp_files/load_file.txt @@ -0,0 +1,1053 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf,len=1052 +page_content='Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing Michael Chesser The University of Adelaide Australia michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='chesser@adelaide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='au Surya Nepal Data61 CSIRO Australia Surya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Nepal@data61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='au Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe The University of Adelaide Australia damith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='ranasinghe@adelaide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='au ABSTRACT Emulation-based fuzzers enable testing binaries without source code, and facilitate testing embedded applications where automated execution on the target hardware architecture is difficult and slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The instrumentation techniques added to extract feedback and guide input mutations towards generating effective test cases is at the core of modern fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' But, modern emulation-based fuzzers have evolved by re-purposing general-purpose emulators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' conse- quently, developing and integrating fuzzing techniques, such as instrumentation methods, are difficult and often added in an ad-hoc manner, specific to an instruction set architecture (ISA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This limits state-of-the-art fuzzing techniques to few ISAs such as x86/x86-64 or ARM/AArch64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' a significant problem for firmware fuzzing of diverse ISAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This study presents our efforts to re-think emulation for fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We design and implement a fuzzing-specific, multi-architecture emulation framework—Icicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We demonstrate the capability to add instrumentation once, in an architecture agnostic manner, with low execution overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We employ Icicle as the emulator for a state-of-the-art ARM firmware fuzzer—Fuzzware—and replicate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Significantly, we demonstrate the availability of new in- strumentation in Icicle enabled the discovery of new bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We demonstrate the fidelity of Icicle and efficacy of architecture ag- nostic instrumentation by discovering LAVA-M benchmark bugs, requiring a known and specific operational capability of instrumen- tation techniques, across a diverse set of instruction set architectures (x86, ARM/AArch64, RISC-V, MIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Further, to demonstrate the effectiveness of Icicle to discover bugs in a currently unsupported architecture in emulation-based fuzzers, we perform a fuzzing cam- paign with real-world firmware binaries using Texas Instruments’ MSP430 RISC ISA and discovered 7 new bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CCS CONCEPTS Software and its engineering → Software testing and debug- ging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' • Security and privacy → Embedded systems security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' KEYWORDS Fuzzing, emulation, embedded systems ACM Reference Format: Michael Chesser, Surya Nepal, and Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In Proceedings of ACM SIGSOFT International Symposium on Software Testing and Anal- ysis (ISSTA 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ACM, New York, NY, USA, 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='org/ XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION Fuzzing is an automated software testing methodology that re- peatedly executes a program with generated inputs and monitors execution for adverse behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Progress in the field has greatly en- hanced the bug discovery capability of modern fuzzers and fuzzing is now widely used in the software development industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In partic- ular, grey-box fuzzing (or feedback-driven) methods have proven to be highly effective at scale and are capable of finding bugs in a diverse set of software [3, 4, 7, 31, 38, 44, 50, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Grey-box fuzzing relies on the ability to add instrumentation to the target binary to obtain feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This feedback allows input generation to be intelligently guided, improving a fuzzer’s ability to discover bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' A simple method to facilitate grey-box fuzzing is for the compiler to inject instrumentation into the source code during compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However, it is often necessary to fuzz binaries where source code is unavailable—binary-only fuzzing—or where the target hardware is not suitable for automating testing and testing is carried out on a host machine with a different instruction set architecture (ISA)—cross-architecture fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For instance, it is extremely chal- lenging to perform rapid execution on devices typically used for Internet of Things (IoT) applications and embedded systems in gen- eral [11, 14, 22, 29, 30, 34, 39, 45, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Consequently, we are forced to use emulators capable of executing binaries built for the target on a more convenient host machine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' exploiting the resource capabili- ties of the host for software bug discovery and triaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Therefore, emulators play a critical role in supporting both binary-only and cross-architecture fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Significantly, emulators enable unpar- alleled control and introspection over program execution, even without source code and access to the original hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Current state-of-practice for emulation-based grey-box fuzzing, driven by its more recent evolution compared to emulators, is to integrate fuzzing instrumentation into existing general-purpose em- ulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' But, this can be challenging because these emulators were not designed to support such modifications [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Consequently, existing emulation-based fuzzers implement instrumentation tech- niques either manually, through direct modifications to the emu- lator [24, 25, 51, 64], or through limited interfaces that are unable to support more advanced instrumentation [29, 39, 40, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Further, the absence of a consistent approach to add new, experimental, instrumentation and the need for domain expertise in emulator de- velopment to evaluate new fuzzing techniques are arguably barriers to developments in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' As a result, the benefits of extensive research efforts to develop state-of-the-art fuzzing techniques can remain limited to a specific ISA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' this is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This study presents our efforts to design and build a new multi-architecture emulation framework explicitly for fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In summary, we make the following contributions: We designed a new multi-architecture emulator, Icicle, for directly supporting emulation-based fuzzing: i) designing for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='13346v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='CR] 31 Jan 2023 ISSTA 2023, 17-21 July, 2023, Seattle, USA Michael Chesser, Surya Nepal, and Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe architecture-agnostic instrumentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ii) employing a decou- pled design, enabling emulation, instrumentation and instruc- tion set architecture (ISA) support to be maintained separately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' and iii) byte level memory-management to better support em- ulating memory in embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We implemented Icicle as a coverage-guided greybox fuzzer by integrating with AFL++ and Fuzzware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We conducted extensive experiments across five diverse ISAs, multiple instrumentation techniques, 21 real-world binaries and a synthetic test program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Notably, we demonstrate: i) the instrumentation requirements of state-of-the-art fuzzing tech- niques are satisfied with a unified instrumentation interface without the need for architecture-specific knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ii) the effectiveness of Icicle by comparing against existing emula- tors for the challenging task of firmware fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Significantly, Icicle, supported by additional architecture-agnostic instru- mentation, uncovered seven previously undiscovered bugs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' and iii) the efficacy of Icicle and its architecture agnostic in- strumentation on a new ISA—we fuzz and discover new bugs in real-world binaries written for Texas Instruments’ MSP430 RISC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Importantly, this ISA is currently not sup- ported by existing emulation-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We open source1 Icicle to facilitate further improvements and advance the field of emulation-based fuzzing in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2 INSTRUMENTATION CHALLENGES IN EMULATION-BASED FUZZING In this section, we present an overview of different instrumentation techniques that are used in modern grey-box fuzzing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Subsequently, we highlight the issues hindering their implemen- tation in existing general-purpose emulators without resorting to direct, architecture-specific, modifications of the emulator that mo- tivate the need of a re-designed emulator for fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1 Instrumentation Techniques Grey-box fuzzers rely on instrumentation techniques to obtain feed- back to enable more effective exploration of a target program which is necessary for uncovering deeper bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Therefore, it is important for fuzzing frameworks to support a diverse set of instrumentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In this section we describe a number of instrumentation techniques developed in previous research efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Code coverage (Branch hit counts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Almost all grey-box fuzzers utilize a form of code coverage for feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Code coverage identi- fies inputs that reach new locations within a program by instrument- ing the target so it notifies the fuzzer when new code is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' It is a proven and effective method that enables fuzzer to incre- mentally discover different parts of the program [8, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Branch hit counts is an approach to code coverage popularized by AFL [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This approach maintains a global map of 8-bit counters that are in- cremented whenever an edge in the program is hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' After execution, the values of each of the counters grouped into one of 8 ranges and if any of the counters contain a value with a unique range, the fuzz input corresponding to the execution is considered novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='com/icicle-emu/icicle Context-sensitive branch coverage [12, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Context-sensitive branch coverage augments branch hit counts by hashing the edge index with the current calling context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This allows the fuzzer to obtain better feedback from branches inside of frequently called functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CmpLog [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For many target binaries, code coverage is insuffi- cient at finding inputs that reach deep parts of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For example, comparisons against large constants (such as Listing 1) are difficult to satisfy with just code coverage, because there is no feedback mechanism that allows for incremental progress towards solving the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' if (tag == 0x31677562) { crash();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' } Listing 1: Example program where crash is hard to reach with traditional code coverage instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' One approach explored in several recent studies [5, 23, 28] is to directly instrument the operands of comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CmpLog is a comparison tracing technique implemented in AFL++ based on Redqeen [5] and Weizz [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CmpLog identifies comparisons within a program, then adds instrumentation that captures the operands of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' After execution, the fuzzing frontend can then scan the input for the operands in order to replace them with their correct value (a process referred to as input-to-state replacement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CompareCov [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Is an alternative approach to solving complex comparisons based on the existing compiler instrumentation tech- nique, LAF-Intel [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CompareCov provides better feedback by instrumenting the program to split comparisons involving large values into comparison between individual bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The split com- parison can subsequently be solved with code coverage instrumen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In addition to instruction-level comparisons, CompareCov also attempts to improve feedback for memory comparison func- tions (memcmp, strcmp and strncmp) by adding instrumentation to update the coverage (bitmap) for every matching byte within the comparison operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='2 Instrumentation Challenges QEMU and Unicorn (forked from QEMU), have become the de facto emulators for fuzzers [13, 14, 16, 22, 29, 30, 39, 40, 45, 51, 52, 58, 59, 65–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Therefore, for the remainder of this paper we will primarily compare Icicle against emulator-level capabilities in QEMU and Unicorn to support fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' QEMU was primarily designed for fast, general-purpose, emula- tion, not for fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' As a consequence, many design decisions differ from those important for fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Crucially, it is difficult to add advanced instrumentation techniques in an architecture agnostic manner, for the following key reasons: QEMU’s intermediate representation for dynamic translation, Tiny Code Generator (TCG) ops, is not designed for direct analysis or manipulation [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Consequently, it is challenging to add instrumentation without making invasive code changes at a very low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing ISSTA 2023, 17-21 July, 2023, Seattle, USA QEMU is monolithic in design, providing little support for extensibility, and assumes that it controls the full life cycle of the emulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' These properties inhibit the maintain- ability of modifications and reduce scalability as it makes it challenging to share state across fuzzing instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Given QEMU’s historical focus on emulation, there is no uni- fied mechanism for adding instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For example, code coverage is implemented by modifying the code-generator to inject code at the start of every basic block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' more complex techniques, such as CmpLog, modify the translation process of individual architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Hence, adding new instrumentation techniques require extensive emulation domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Unicorn is a fork of QEMU designed for flexible and modular CPU emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Unicorn extracts the CPU emulation component from QEMU, configures it to always use the software MMU im- plementation, and introduces hooks, functions that are called in response to selected emulator events, such as memory accesses and breakpoints [39, 40, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Unicorn’s function hooking API, enables fuzzers to inject func- tions calls to observe the CPU state, which can be used to implement some instrumentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However, the observable state is architecture specific and Unicorn provides no support for ana- lyzing the code semantics, making more advanced instrumentation difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The maintainers of the Unicorn project have also reported that it is difficult to keep Unicorn up-to-date with improvements because of QEMU’s monolithic design [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Therefore, we are motivated to build a new multi-architecture emulation framework explicitly designed for fuzzing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' with the ability to support sophisticated instrumentation methods in an architecture-agnostic manner to enable fast emulation-based fuzzing of binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 3 ICICLE DESIGN AND IMPLEMENTATION We provide an overview of our fuzzing specific multi-architecture emulation framework in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To enable architecture agonostic fuzzing we use a portable intermediate representation (IR) that is suitable for both emulation and program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Translation of the guest ISA to the portable IR is achieved using processor specifications that are external to the emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This ensures that architecture-specific details are kept decoupled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' enabling fixes for specification bugs and the addition of new architectures to be im- plemented with minimal changes to the core emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Further, in contrast to existing emulation-based fuzzing frameworks, we define new instrumentation application programming interfaces (APIs) that enable instrumentation to exist entirely outside of the emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This facilitates researchers both in developing new instru- mentation techniques for emulation-based fuzzers without domain expertise in emulator development,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' and the immediate availability (SLEIGH specifications) x86-64 ARM MIPS 1 JIT invalidation ICICLE Instruction Optimizer Decoder Lifter Guest ISA Instructions P-code Block translator JIT cache Snapshot Store JIT function Block map P-code block Modified blocks New block event JIT Compiler Code generation (Cranelift) Software MMU P-code Emulator Environment Emulator (Linux/MSP430 MCU) Loader (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='elf/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='bin/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='ihex) Virtual Registers Dispatcher TLB Tracer Plugins 3 4 5 6 7 8 Code coverage CmpLog CompareCov 2 SLEIGH runtime Figure 1: Overview of the core components in our fuzzing specific multi-architecture emulation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The instrumen- tation workflow is as follows: 1 On initialization the emulator loads the appropriate SLEIGH processor specification, config- uring the SLEIGH runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2 One or more Tracer Plugins are registered with the emulator to support the instrumentation needs of the fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 3 Once configured, the emulator loads the target binary into memory and starts execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4 During execution, whenever the emulator attempts to execute a new instruction, the dispatcher initiates the translation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 5 Each guest instruction is then translated to P-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 6 P-code operations are grouped into a block, and the block is stored in a global block map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 7 Tracer Plugins are notified to allow them to analyze the new block and modify any block in the block map required for instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 8 Any new or modified blocks are then compiled to native code using the JIT compiler and the emulator resumes execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Notably we implemented the components in the grey area and Icicle is more than 32,000 lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ISSTA 2023, 17-21 July, 2023, Seattle, USA Michael Chesser, Surya Nepal, and Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe of these techniques across ISAs in a design-build-and-test once-only paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1 Fuzzing Specific Emulator Design Icicle supports fast, multi-architecture, CPU emulation through portable dynamic translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' First, guest ISA instructions are trans- lated to an intermediate representation (IR) called P-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' P-code is then just-in-time (JIT) compiled to the host architecture allowing for efficient execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle performs translation to P-code through the use of a SLEIGH2 processor specification for the guest ISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' SLEIGH is a domain- specific language (DSL) that describes how to decode and translate the semantics of machine code into P-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We chose SLEIGH as the basis of Icicle’s CPU emulation for the following reasons: Broad architecture support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We leverage the diverse set of SLEIGH specifications that have already been created as part of the open-source Ghidra framework [60] (over 45 processor kinds are supported).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This enables Icicle to emulate a wide range of architectures, including architectures unsupported by other emulation frameworks like QEMU, with significantly reduced effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Designed for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Unlike IRs used in other emulator frame- works, P-code was explicitly designed to support program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For example, P-code maintains hints for call and return operations even though such hints are unnecessary for emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This makes it better suited for performing the code analysis tasks required for advanced instrumentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Suitable for emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' P-code consists of small set of oper- ations that can be efficiently executed by the host ISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For example, P-code avoids the use of bit-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This allows for fast emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Decoupling and ease of maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle is able to use the original SLEIGH specifications written for Ghidra without any modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This allows any improvements or fixes made to a specification to be immediately usable by Icicle without any changes to the core emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle implements a custom P-code emulator3 consisting of sev- eral components, a custom SLEIGH runtime, a JIT-based execution engine and a software memory-management unit (MMU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The SLEIGH runtime handles loading the appropriate SLEIGH specifi- cation for the guest architecture, assigning a mapping from guest registers to virtual P-code registers, and then decoding and translat- ing ISA-specific machine code to P-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Unlike Ghidra’s SLEIGH runtime, Icicle’s runtime assigns sequential IDs to virtual regis- ters, allowing them to be managed in a dense array, improving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We also implement a lightweight P-code optimization pass that performs constant evaluation and dead-code elimination, significantly reducing the amount of P-code operations when val- ues are known at translation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle’s JIT-based execution engine, then groups P-code operations in blocks and compiles them to native code using Cranelift [1], an open-source low-level code 2The name SLEIGH was originally derived from SLED (Specification Language for Encoding and Decoding), which also influenced the name of our emulator: Icicle 3Ghidra contains a limited P-code emulator and has been used for micro-fuzzing [19], but is unable to satisfy the needs of a full modern fuzzing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For instance, Ghidra’s P-code emulator is interpreter-based hindering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' generation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Cranelift provides register allocation, in- struction legalization, and additional optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Later, during a recompilation step, multiple blocks are compiled as part of a single compilation unit, enabling additional optimizations that improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Notably, unlike existing emulators, Icicle does not discard the P-code representation of each block after JIT compi- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This can significantly aid any analysis used for complex instrumentation, at the cost of some additional memory overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' When the memory layout of the guest is incompatible with the host, it is necessary for the emulator to handle the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Therefore, Icicle uses a software MMU to handle guest memory accesses (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The software MMU maintains virtual mapping table that allows guest memory and memory mapped IO (MMIO) to be mapped in the emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The mapping table represented as a range-map (implemented with B-Tree), allowing for byte-level precision at the cost of more expensive lookups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To improve effi- ciency, we cache translated addresses in a lookup table referred to as a translation lookaside buffer (TLB), named after its hardware analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The JIT compiled code, can directly access guest memory using the TLB allowing for fast execution in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Whenever, an address not in the TLB, is encountered, the JIT calls a runtime helper that handles the access and caches the translated address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To retain byte-level mapping, Icicle maintains a permission byte for each physical byte which is checked by the JIT on access, similar to approaches used in prior work [21, 24, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Both Unicorn and QEMU (when running in full-system mode) also implement a soft- ware MMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However, they both require memory to be mapped in page-sized (4 KB) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The added flexibility of the byte-level mapping in Icicle allows more accurate emulation of embedded system memory, and can be used to enable better bug detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Virtual mapping TLB (direct mapped) Physical memory pages Data 0 Perm 0 Data 1 Perm 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Tag Pointer Real memory Mapped memory (unallocated) I/O handler (unmapped) I/O Callbacks JIT access Runtime access Figure 2: Overview of the byte-level software memory- management unit (MMU) implemented in Icicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In addition to CPU emulation, most binaries interact with exter- nal resources such as file systems, hardware, and other software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle is designed to be flexible and extensible through the use of pluggable environment emulators (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To demonstrate the functionality of the system, we have implemented environments to allow comparisons against existing emulation-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The current implementation supports fuzzing Linux userspace binaries Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing ISSTA 2023, 17-21 July, 2023, Seattle, USA by emulating a subset of system calls, supports fuzzing several MSP430 MCUs ISAs, and supports embedded ARM binaries using Fuzzware [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='2 Architecture-Agnostic Instrumentation To implement arbitrarily complex fuzzing instrumentation requires: i) the ability to analyse the semantics of a target program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ii) an effi- cient mechanism to capture runtime information about the running program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' iii) a way of sharing the captured information with the fuzzing frontend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Additionally, to be effective in a fuzzing context, these requirements must be supported in a manner that has low performance overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle supports these requirements through a set of APIs added to the emulator, we refer to instrumentation utilizing these APIs as Tracer Plugins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' These APIs enable: Direct access to the architecture-agnostic P-code representation of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Plugins are able register a callback function to be called whenever the emulator translates a new block to P-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The callback function is provided with the full code- cache including the newly translated block, satisfying the first requirement enabling architecture agnostic code analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Inline code-injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Plugins can inject additional P-code op- erations into any block enabling inline instrumentation to be supported in an architecture agonstic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Any modified blocks are invalided by Icicle and re-compiled by the JIT the next time they are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Registry of JIT and fuzzer accessible shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' During ini- tialization, plugins are able to register storage locations with the emulator, which can later be manipulated with P-code op- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Additionally, Icicle allows plugins to define custom P-code registers, these custom registers are treated the same as guest registers for the purpose of register allocation during JIT code-generation, which can allow for more efficient instru- mentation in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This enables data to be efficiently saved by injected instrumentation and analyzed as part of the fuzzing loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To illustrate expressiveness Icicle’s instrumentation method and it ability to support architecture-agnostic instrumentation, we discuss the implementation of the four techniques discussed in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1 in Icicle and compare them to implementations in other emulation-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Branch hit counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In Icicle, branch hit counts are implemented by a Tracer Plugin that does the following: during initialization, it registers the location of coverage bitmap with the emulator and de- fines a custom register to store the previous program location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' When a new block is translated, the plugin injects code at the start of the block that computes a hash of (current_location, previous_location), which is then used as an index for updating the coverage bitmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since the instrumentation is implemented using P-code injections, the JIT is able to generate native code that updates the coverage bitmap without resorting to a function call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In addition to branch hit counts, Icicle also implements both block-only coverage and edge coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Existing emulators are also able to add branch hit count instru- mentation in an architecture independent manner by injecting code when new translation blocks are created, which is common across architectures in QEMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However, AFL++’s implementation in both QEMU and Unicorn makes direct modifications the emulator (al- though the changes are relatively minor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Notably, in AFL++’s Uni- corn mode, branch coverage instrumentation is not implemented using Unicorn’s hooking API, since the instrumentation is highly performance sensitive and the hooking API imposes additional overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Context-sensitive branch coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle implements context- sensitive branch coverage with a Tracer Plugin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This plugin defines a custom register to store the context, then when a new block ending with a CALL is translated, the generates a random value to use as context for the current location and XORs it the context register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In the block after the call, the instrumentation is injected to clear the added context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The branch hit count plugin is then modified to use the context value by using it as part of computing the index into the coverage map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since the CALL hint is part of P-code representation, it allows us to write a portable implementation that works across architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Context-sensitive coverage was first implemented in Angora[12] using compiler-based instrumentation, and in afl-sensitive [62] for binary-only instrumentation using QEMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' afl-sensitive’s implemen- tation modifies QEMU’s x86 translation layer to add instrumenta- tion that updates the calling context on call and ret instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since afl-sensitive instruments x86 specific instructions it is not portable to other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CmpLog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' There are two parts to CmpLog, first relevant comparison operations must be identified, and second, the operands of each comparison must be copied to a fuzzer accessible location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Inspired by the success of Datalog for program analysis tasks [26, 57], we implement a comparison finding algorithm as a set of Data- log rules in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since the rules are defined in terms of P-code operations, it allows Icicle support CmpLog for any architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % x is an copy of the destination of an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' copy(x, x) :- op(x, _, _, _).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % b = a if it is the destination of a copy-like op with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' copy(a, b) :- op(b, "COPY", a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' copy(a, b) :- op(b, "ZXT", a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % b = a if x = a and b = x copy(a, b) :- copy(a, x), copy(x, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % Identify p-code operations corresponding to comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' cmp("==", cond, a, b) :- op(cond, "==", a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' cmp("!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='=", cond, a, b) :- op(cond, "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='=", a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % `(a - b) [cmp] 0` => `a [cmp] b` (subtract and compare with zero) cmp(op, cond, a, b) :- op(cond, "-", a, b), cmp(op, cond, x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % `!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' (a [inv(op)] b)` => `a [op] b` (inverted comparison) cmp("!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='=", cond, x, y) :- op(notc, "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ", cond), cmp("==", notc, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' cmp("==", cond, x, y) :- op(notc, "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ", cond), cmp("!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='=", notc, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' % Output comparisons that flow into the branch condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' output(op, a, b) :- cmp(op, cond, a, b), copy(cond, x), branch(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Listing 2: Datalog rules for finding comparison operands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The list of p-code operations to analyse, and the branch exit condition are provided as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In contrast, existing CmpLog implementations require identi- fying architecture specific instructions in order to identify com- parisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For example, on x86, AFL++’s instruments CMP and SUB instructions, by modifying QEMU’s translation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This has two main issues: 1) since the instrumentation looks for specific instruc- tions, a separate implementation is required for each architecture, ISSTA 2023, 17-21 July, 2023, Seattle, USA Michael Chesser, Surya Nepal, and Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe 2) it can result in excessive instrumentation, for example most SUB operations on x86 are not used for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CmpLog is not supported in Unicorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CompareCov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In Icicle, integer comparisons are identified using the same algorithm as CmpLog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Once identified, Icicle injects code that writes to the coverage bitmap for each matching byte before the original comparison operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For memory comparisons functions, Icicle searches for the target functions in the program’s symbol table and injects instrumentation when a block calling the target function is translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This allows Icicle’s instrumentation to be used for statically linked binaries including firmware (as long as the symbol table has not been stripped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In contrast, AFL++’s implementation for integer comparisons requires identifying architecture specific comparison instructions, like CmpLog instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Further, for memory comparison functions, it relies on the dynamic linker to replace the original comparison functions with instrumented versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This approach is unable to support instrumenting statically linked firmware binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Each instrumentation technique is implemented targeting P-code enabling it to support any ISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' And, as an added benefit, only knowledge of P-code is adequate for devel- oping new instrumentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='3 Fuzzing Frontend Integration Modern grey-box fuzzing frameworks consists of two main compo- nents: the frontend which handles input generation, input sched- uling, hang detection and crash deduplication, and the backend which manages program execution, crash monitoring, and instru- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Emulation-based fuzzers utilize emulators as the fuzzing backend allowing for binary-only and cross-architecture fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ici- cle is a new fuzzing backend, therefore, we make our emulator compatible with an existing fuzzing framework: AFL++ [25] to avoid implementing a new frontend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' AFL++ is a state-of-the-art fuzzing framework derived from the well-known American Fuzzy Lop (AFL) [64] project, with general improvements, and support for additional fuzzing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle integrates with AFL++ using the forksever interface also used by AFL++’s QEMU-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4 EVALUATION Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Unless otherwise specified, all experiments were carried out with AFL++ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='01a as the fuzzing frontend on an AMD Ryzen Threadripper 3990X restricted to a single core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' All AFL++ settings were kept as default, except to enable instrumentation as needed and to adjust the timeout for hang detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We design our experimental regime to answer five specific questions articulated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1 Is Icicle’s instrumentation portable across diverse ISAs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To ensure that architecture-agnostic instrumentation implemented in our emulator is operational across a range of architectures, we designed a test program, shown in Listing 3, that consists of 5 synthetic bugs designed to test specific instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=" void test_instrumentation(char* buf) { // (1) comparison against a single byte in the input if (buf[0] == '%') { crash(1);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' } // (2) Multiple comparison against single bytes of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=" if (buf[0] == 'i' && buf[1] == 'x' && buf[2] == 'S' && buf[3] == 'D') { crash(2);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' } // (3) A single comparison against multiple input bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' if (*(u32*)buf == *(u32*)"wzfc") { crash(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' } // (4) A multi-byte comparison across a function call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' if (compare(buf, "dGlIHF1W") == 0) { crash(4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' } // (5) Saturate coverage then compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' saturate_compare2_cov();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' u32 tmp = *((u32*)buf) ^ 0x46092d5f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' if (compare2(tmp, 0x7451496b)) { crash(5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' } } Listing 3: Test program used for evaluating instrumenta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We evaluate the portability of Icicle’s instrumentation by fuzzing the test program compiled for 5 different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For archi- tectures with Linux support, we configure the program to read the input from stdin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For MSP430, the program reads from a peripheral mapped to the fuzzing input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' After compiling the binary for each architecture, we manually verified that the machine code of out- put binary behaves as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' As a baseline we compare against AFL++’s QEMU-mode when instrumentation is supported for the guest architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For each fuzzing configuration, we perform 20 trials for a maximum of 10 minutes starting with an uninformed seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The results from this experiment are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Both Bug1 and Bug2 are discoverable with code coverage alone, so are found by all fuzzing configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Bug3 requires additional instrumentation to be found so can only be found when one of the two comparison instrumentation techniques is enabled, except for the MSP430 binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' With CmpLog, the fuzzer can find a solution via an input-to-state mutation directly replacing the incorrect value with a correct one, with CompareCov enabled, the comparison is “split” into byte-level comparisons, and the fuzzer observes incre- mental coverage feedback similar to Bug2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since MSP430 is a 16-bit architecture, the compiler splits the 32-bit comparison into two 16- bit comparisons allowing the fuzzer to eventually find the crashing input for Bug3 without additional instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Bug4 evaluates the fuzzers ability to solve memory comparison functions so is only discovered when CmpLog instrumentation enabled, which gener- ally finds the crashing input within seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CompareCov fails to find the bug, since compare is not a standard comparison function and is therefore not instrumented by CompareCov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CmpLog is only partially implemented for QEMU on AArch64 (function calls are not instrumented) so fails to find Bug4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Bug5 tests the fuzzer’s ability to find a bug in a function where code coverage is saturated by a pre- vious call, so is only found when context sensitive branch coverage is enabled, which only Icicle supports on all architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing ISSTA 2023, 17-21 July, 2023, Seattle, USA Table 1: Results from different fuzzing instrumentation configurations for the test program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ✓ denotes the bug ID was found at least once within 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Each test was repeated 20 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Shaded grey areas are due to: i) unsupported fuzzing instru- mentation for MIPS and RISC-V in QEMU emulation with AFL++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' and ii) MSP430 ISA being unsupported in QEMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='x86-64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='AArch64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='MIPS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='RISC-V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='MSP430 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Fuzzer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Instrumentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ Icicle (ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Cov+Context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ✓ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ QEMU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Cov+CompareCov ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='✓ Summary The test program binaries for five different ISAs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='provide empirical evidence that the architecture agnostic in- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='strumentation implementation of the different instrumenta- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='tion techniques in Icicle is both effective and portable across ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='2 Is architecture-agnostic instrumentation as effective as existing architecture-specific implementations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' LAVA-M [20] is a widely used set of binaries for evaluating and benchmarking fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' It consists of four binaries from GNU core- utils [27] each injected with synthetic bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' While the injected bugs are not representative of typical real-world vulnerabilities [35], pre- vious work has demonstrated that these bugs are difficult to find with code coverage only, however can be found with by instru- menting comparison operations [5, 12, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This naturally lends itself to assessing Icicle’s architecture-agnostic implementation of CmpLog and CompareCov instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Using AFL++ as the frontend, we evaluate the bug discovery capability of Icicle across four different ISAs (x86, AArch64, MIPS, and RISC-V), the first two of which we compare against QEMU4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We also evaluate both QEMU and Icicle on x86 with code coverage only as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For each of the injected bugs, a unique ID is written to stdout whenever the bug is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Therefore, we can verify each crash by running the x86 version of the binary on the host machine then checking for unique bug IDs in stdout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We perform 5 trials for each fuzzing configuration running for 12 hours each, starting with the same two initial seeds as Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The results of the benchmark are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' With code coverage alone almost no bugs are found by either emulator in any of the binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Both comparison instrumentation techniques allow most bugs to be found, with CmpLog finding bugs significantly faster than CompareCov in several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle’s re- sults closely match QEMU results for both AArch64 and x86, which supports our claim that Icicle’s instrumentation is as effective as the architecture-specific approach employed by AFL++’s QEMU- mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' On the two additional architectures tested with Icicle both instrumentation techniques continue to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However, the 4We compare with QEMU not Unicorn, since Unicorn cannot directly execute Linux binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Additionally, since neither CmpLog nor CompareCov instrumentation are supported in AFL++’s QEMU-mode for RISC-V and MIPS, we only evaluate these architectures with Icicle results for the MIPS version of uniq are slightly worse, this is caused by differences in the memory layout (MIPS uses a 32-bit ad- dress space, while the other architectures are 64-bit), which causes issues when replaying the crashing input on the x86 host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The differences in the number of crashes found for who binary across architecture is caused caused by platform specific behaviour in the program itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The fuzz input is parsed as a utmpx structure, however the layout of the fields within the structure is different across architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This can cause certain bugs to become un- reachable, and also causes issues when we attempt to replay the crashing inputs on the x86 version of the binary in order to verify the crash IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Additionally, the binary frequently crashes before a bug ID is flushed to stdout (caused by internal buffering), which prevents us from obtaining the bug ID from the original execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Notably, all bugs reported and discovered are those reproduced on both the guest architecture and the host (x86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This is additional evidence of the importance of binary-only and cross-architecture fuzzing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' even when source code is available, program behaviour can differ on between architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Summary Discovering LAVA-M benchmark bugs require a specific operational capability from instrumentation tech- niques to solve comparison operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' namely CompCov or CompLog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle’s results closely match QEMU results for both AArch64 and x86, supporting our claim that Icicle’s instrumentation is as effective as the architecture-specific approach employed by AFL++’s QEMU-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' On MIPS and RISC-V architectures (where AFL++’s QEMU-mode does not support the necessary instrumentation) both instrumentation techniques tested with Icicle continue to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='3 Can Icicle be used to implement and enhance state-of-the-art firmware fuzzing techniques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Fuzzware [52] is a recent state-of-the-art fuzzing framework for analyzing ARM firmware binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Fuzzware extends Unicorn to in- strument and execute ARM firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We replace Unicorn with Ici- cle to evaluate Icicle’s ability to support state-of-the-art firmware fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We then tested our modified version (Fuzzware-Icicle) by attempting to reproduce Fuzzware’s results on the 10 binaries used in the P2IM [22] firmware set as they were evaluated extensively ISSTA 2023, 17-21 July, 2023, Seattle, USA Michael Chesser, Surya Nepal, and Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe by [22, 67] and Fuzzware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Importantly, since Icicle’s instrumenta- tion is portable we are able to support additional instrumentation when fuzzing ARM firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In particular, we perform additional tests with CompareCov instrumentation enabled to allow for better comparison solving5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We followed the same experimental setup for Fuzzware as described in the original paper (we used the same number of trials, seeds and run time duration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We were able to successfully rediscover all 16 of the bugs found by Fuzzware, and additionally, with CompareCov enable, Fuzzware- Icicle was able to find an additional bug in the Console binary not reported by any prior work6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' As part of the rtc settime command, the firmware reads a date from the user in the form YYYY-MM-DD HH:MM:SS without checking whether the parsed date is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This causes an out-of-bounds access when the name of the month is resolved using a lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since reaching this bug requires first solving a string comparison to reach the rtc handler, then solving a second string comparison for the settime subcommand, we believe the added instrumentation was critical to finding this bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle also found an additional crash in the Soldering Iron bi- nary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' At high temperatures, rendering the heat indicator causes the buffer allocated for the LCD screen to overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However, after fur- ther analysis we discovered the maximum temperature is restricted in software, indicating that the bug is a false-positive caused by Fuzzware’s peripheral modelling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In addition to reproducing the bugs, we also investigated whether Fuzzware-Icicle is able to maintain the same level of block cov- erage as the original implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For almost all of the evaluated binaries we achieve almost identical block coverage to Fuzzware, with some small differences 5We did not test with CmpLog, since effective use of the instrumentation requires additional integration with fuzzing frontend, unsupported by Fuzzware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 6Crashing inputs for each of the discovered bugs are available in our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' of which we manually investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' With CompareCov enabled, Fuzzware-Icicle achieves higher coverage in two of the binaries: Console, and Steering Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The higher coverage in Console corresponds to reaching different command handlers that are dis- patched based on string comparisons, including the sub-commands of the rtc handler that contains the bug discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Simi- larly, Steering Control contains two commands, that are triggered when the matching string is read by the firmware ("steer", and "motor").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' CompareCov enables Icicle to generate inputs contain- ing the command strings, and thus is able to reach additional code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The discrepancies in the Gateway and Soldering Iron binaries are caused by high variance between fuzzing runs, running additional trials would likely remove any discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Summary Icicle is a robust emulator capable of supporting the current state-of-the-art ARM firmware fuzzer, Fuzzware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We discovered all 16 known bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle provides a direct substitute for Unicorn with the added advantage of additional, architecture agnostic instrumentation shown to be effective at improving coverage and discovering 2 new bugs not reported by prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='4 Can Icicle discover bugs in real world binaries in an ISA currently not supported by emulation-based fuzzers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To demonstrate the architecture-independent benefits of our proto- type emulator, we investigate fuzzing firmware written for MSP430 microcontrollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' FiE [17], is the only prior study that attempted to find bugs in MSP430 firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' FiE requires C source-code and therefore does not support manually written assembly code (which is common in larger firmware),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' and is incapable binary-only x86-64 (Cov+CmpLog) (Cov+CompCov) (Cov+CmpLog) AArch64 (Cov+CompCov) MIPS (Cov+CmpLog) (Cov+CompCov) RISC-V (Cov+CmpLog) (Cov+CompCov) md5sum uniq who 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 0 20 40 0 20 40 60 0 10 20 30 0 1000 2000 Duration (h) Bugs found ICICLE QEMU base64 Figure 3: LAVA-M bugs found over time in each binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The solid line represents the median number of bugs found, the shaded area represents the min/max coverage across all trials, and the black dotted lines represent the number of bugs listed in the LAVA-M paper (Note: it is well known that it is possible to trigger additional bugs other than specified in the original paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing ISSTA 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 17-21 July,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' USA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='PLC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Reflow Oven ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Soldering Iron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Steering Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='CNC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Console ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Drone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Gateway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Heat Press ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='16 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Duration (h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='#Blocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='ICICLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Unicorn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='ICICLE (CompareCov) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='Figure 4: Block coverage over time for ARM firmware using the two different emulators and CompareCov instrumentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='supported in Icicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The solid line represents the median coverage of 5 runs, and the shaded area represents min/max coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Table 2: Discovered vulnerabilities in MSP430 ISA binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Firmware Bug description (PoCs & stack traces on GitHub) Goodwatch Incorrect comparison when writing to log buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Goodwatch Buffer overflow when handling zero length packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Goodwatch Stack overflow in RNG generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Goodwatch Out-of-bounds access in OOK keypress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Goodwatch Out-of-bounds access in Stopwatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' H4_PacketProtocol Unchecked Interface Index in Get Descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' H4_PacketProtocol Buffer overflow in Set Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Further, MSP430 firmware is not supported by any exist- ing emulation-based fuzzing framework7, and therefore presents a compelling use case for fuzzing with Icicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Similar to existing monolithic firmware fuzzing approaches [22, 52], we handle peripheral accesses for MSP430 firmware by read- ing them from the fuzzer input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We found this highly effective at finding bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We selected 3 different firmware to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' First, inspired by FiE, we evaluated the USB SDK provided as part of TI’s MSP430 USB Developers Package8 using example programs provided as part of the development package (H4_PacketProtocol) as a harness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Second, we compiled an unmodified version of the Goodwatch [61] firmware9, a hardware and firmware replacement for Casio calculator watches based on the CC430 MCU (a MSP430 CPU with an integrated RF transceiver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Additionally, we investi- gated Icicle’s ability to test closed-source firmware by extracting the firmware off a commercial medical device, a Polar heart rate tracker, containing a MSP430F2132 MCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The block coverage results are summarized Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' After triag- ing the results, we identified two unique bugs H4_PacketProtocol and 5 unique bugs in the Goodwatch firmware and 3 additional crashes related to debugging features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' While no bugs were discov- ered for the Polar heart rate tracker, the fuzzer reached almost all 7[36] is a fork of QEMU adding MSP430 support, however is outdated, not integrated with any fuzzing framework and does not support MSP430 CPUX extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 8https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='com/tool/MSP430USBDEVPACK version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='03 9commit: c8859f845fccf56585a127059b1d1b825b381673 Table 3: Block coverage (#BB) for MSP430 binaries with and without CmpLog instrumentation enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Avg represents the median coverage achieved after 24 hours in 5 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Firmware #BB total Instrumentation #BB min #BB avg #BB max Goodwatch 3263 Cov 2336 2362 2441 Cov+CmpLog 2438 2503 2526 H4 Packet Protocol 925 Cov 819 821 891 Cov+CmpLog 813 910 914 Heart Rate Tracker 744 Cov 679 679 716 Cov+CmpLog 680 717 718 blocks in the firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The bugs discovered by Icicle are summa- rized in Table 2, and for each bug discovered we provide input files and a detailed crash analysis in our GitHub repository10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Summary MSP430 firmware fuzzing is not supported by ex- isting emulation-based fuzzing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Case studies with Icicle and its suite of architecture agnostic instrumentation discovered seven undiscovered software bugs in two (USB SDK–H4_PacketProtocol, and Goodwatch) of the three tested MSP430 binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='5 How does Icicle perform in fuzz testing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In the development of Icicle, we made efforts to ensure that Icicle has good performance in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' We compared fuzz test execution speed of Icicle with Unicorn (emulator) employed by the sate-of- the-art fuzzer, Fuzzware, on the P2IM dataset [22] and summarise the results in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Summary Icicle has approximately the same performance as Unicorn for fuzzing monolithic firmware binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='com/icicle-emu/icicle/crash-analysis ISSTA 2023, 17-21 July, 2023, Seattle, USA Michael Chesser, Surya Nepal, and Damith C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Ranasinghe PLC Reflow Oven Robot Soldering Iron Steering Control CNC Console Drone Gateway Heat Press 0 30 60 90 120 0 20 40 60 80 0 100 200 300 0 100 200 0 50 100 150 0 100 200 0 25 50 75 100 0 10 20 30 0 50 100 150 200 250 0 50 100 150 Execs/second ICICLE Unicorn Figure 5: Icicle and Unicorn performance comparison when integrated with the state-of-the-art fuzzer, Fuzzware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 5 DISCUSSION AND LIMITATIONS Although we have taken the first steps to re-think and re-design an emulation framework to directly support fuzzing requirements, and instrumentation development and testing, the current implemen- tation is not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The released emulator prototype was primarily designed for CPU ISA emulation, similar to the goals of the Unicorn project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' As a result, Linux emulation is minimal, and more complex hardware emulation required for full-system emulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=', page-table emulation) is not currently supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='1 Emulator Correctness In emulation-based fuzzing, since the program not executed on the original hardware, there is a risk that any crashes discovered could be caused by emulation issues, not bugs in the target program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' To reduce the chance of emulation bugs in Icicle, first, we employ a differential testing strategy, similar to other widely used approaches for testing CPU emulators [2, 32, 41–43, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Second, we manually investigated any crashes discovered in benchmark evaluations and ensure they are caused by program bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='2 Performance In the development of Icicle, while we made efforts to ensure that Icicle has good performance in general, there are a number of addi- tional optimizations possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' The current implementation of Icicle has demonstrably similar performance to Unicorn (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Al- though a direct performance comparison against QEMU is desirable, it is more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Because, Icicle implements a forkserver similar to AFL++’s persistent mode, however we run AFL++’s QEMU-mode without this feature since (currently) persistent mode requires a significant amount of manual effort to set up for each binary (no- tably, Icicle’s implementation is automatic for Linux binaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This results in Icicle performing significantly faster for small binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle always translates memory accesses in software (like Uni- corn) while AFL++’s QEMU-mode can utilize hardware address translation when running Linux user-space binaries on a Linux host resulting in a significant speedup for QEMU for larger Linux binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 6 RELATED WORK Improving emulation-based fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' There has been some ef- fort in improving QEMU and Unicorn for fuzzing, including, im- proving runtime performance [6, 25], enabling support for full- system emulation of Linux-based firmware [11, 15, 59, 65, 66], and extending the emulator to support additional analysis such as taint tracking and symbolic execution [51, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Binary-only fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Without access to source-code it is chal- lenging to use fuzzing techniques that rely on instrumentation, since the simplest approach using compiler-based code injection, is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Fuzzers that support targets without source-code are known binary-only fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Emulation-based approaches are one solution, however there are several other alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Virtualisation/hardware-assisted approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=', kAFL [54], and NyX [53, 55]) use a variety of hardware features to implement fuzzing instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Since they require additional hardware support some instrumentation cannot be easily implemented, and firmware fuzzing is not supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Static rewriters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=', Retrowrite [18], Datalog Disassembly [26], Zafl [46]) disassemble a binary, inject instrumentation, then reassemble the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This can enable close to compiler-level instrumentation performance, however the com- plexity involved in the rewriting process often results in correctness issues, typically firmware binaries are not well supported, and static rewriting cannot be used cross-architecture fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' While, dy- namic instrumentation tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=', DynamoRIO [9], PIN [37], CMU BAP [10], Valgrind [47]), share significant similarities to emulation- based approaches, they are more restrictive than full emulators and are unable to support firmware fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Embedded system fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Fuzzing embedded systems and IoT devices is difficult because we cannot avoid dealing with hardware and peripheral interactions since it might represent a majority of the code we are trying to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' As a result, emulation-based fuzzers need to support more than just CPU emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Past work has extended either QEMU or Unicorn to support firmware fuzzing through, hardware-in-the-loop approaches [16], peripheral model- ing [17, 22, 30, 67], or emulating the hardware abstraction layer [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' More recently, Fuzzware [52], outperformed prior firmware fuzzing approaches by automatically generating peripherals models using local symbolic execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Notably, aside from FiE (which is source- based and only targets MSP430), all these approaches only evaluate firmware written for the ARM architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle makes it eas- ier to fuzzing multiple architectures, which we hope will assist in increasing the architecture diversity in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 7 CONCLUSION Emulation-based fuzzing techniques, supported by effective instru- mentation, are highly flexible and are the only method for cross- architecture fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' For historical reasons, emulators used in ex- isting emulation-based fuzzing frameworks were not designed for fuzzing and has made it difficult to meet fuzzing specific needs such as implementing advanced instrumentation techniques supporting a design-build-and-test once-only paradigm across multiple ISAs, and implementing fuzzing specific optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Icicle: A Re-Designed Emulator for Grey-Box Firmware Fuzzing ISSTA 2023, 17-21 July, 2023, Seattle, USA We designed and implemented a new multi-architecture emu- lation framework for fuzzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Within our framework, we imple- mented four different architecture agnostic instrumentation tech- niques and demonstrated that a single architecture-independent implementation is effective across multiple architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Our em- ulation platform is extremely flexible, supporting a wide range of ISAs, especially significant in fuzzing firmware in embedded systems and IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' This was demonstrated by discovering 7 new bugs in ARM firmware by integrating with the state-of-the-art ARM firmware fuzzer, Fuzzware and fuzzing firmware for MSP430 ISAs—an unsupported target by existing emulation-based fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT We uploaded artifacts to https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='com/icicle-emu/icicle, in- cluding source code, a README guide for users, PoCs and stack traces for newly discovered bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The work has been supported by the Cyber Security Research Cen- tre Limited whose activities are partially funded by the Australian Government’s Cooperative Research Centres Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' REFERENCES [1] Bytecode Alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} 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Real-World Embedded Systems Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In 27th USENIX Security Symposium (USENIX Security 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 309–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [17] Drew Davidson, Benjamin Moench, Thomas Ristenpart, and Somesh Jha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' FIE on firmware: Finding vulnerabilities in embedded systems using symbolic execution.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [21] Brandon Falk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Vectorized Emulation: MMU Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' https:// gamozolabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='io/fuzzing/2018/11/19/vectorized_emulation_mmu.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Fuzzing Binaries for Memory Safety Errors with QASan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In 2020 IEEE Secure Development Conference (SecDev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 23–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [25] Andrea Fioraldi, Dominik Maier, Heiko Eißfeldt, and Marc Heuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' AFL++: Combining Incremental Steps of Fuzzing Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In 14th USENIX Workshop on Offensive Technologies (WOOT 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' USENIX Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [26] Antonio Flores-Montoya and Eric Schulte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Datalog Disassembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In 29th USENIX Security Symposium (USENIX Security 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' USENIX Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [27] Free Software Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' GNU core utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' https://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In 29th USENIX Security Symposium (USENIX Security 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [51] Sebastian Poeplau and Aurélien Francillon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' SymQEMU: Compilation-based symbolic execution for binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In Network and Distributed System Security Symposium (NDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Efficient greybox fuzzing of applications in Linux-based IoT devices via enhanced user-mode emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' In Proceedings of the 31st ACM SIGSOFT International Symposium on Software Testing and Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 417–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' [67] Wei Zhou, Le Guan, Peng Liu, and Yuqing Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' Automatic Firmware Emulation through Invalidity-guided Knowledge Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content='. In USENIX Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} +page_content=' 2007–2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFQT4oBgHgl3EQfhjY0/content/2301.13346v1.pdf'} diff --git a/cdFIT4oBgHgl3EQfnSu4/content/tmp_files/2301.11313v1.pdf.txt b/cdFIT4oBgHgl3EQfnSu4/content/tmp_files/2301.11313v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c6dbe402452c753769957e7b453982ea1ba2259 --- /dev/null +++ b/cdFIT4oBgHgl3EQfnSu4/content/tmp_files/2301.11313v1.pdf.txt @@ -0,0 +1,1969 @@ +1 +Distributed Optimization Methods for Multi-Robot +Systems: Part I — A Tutorial +Ola Shorinwa,1 Trevor Halsted,1 Javier Yu,2 Mac Schwager2 +Abstract—Distributed optimization provides a framework for +deriving distributed algorithms for a variety of multi-robot +problems. This tutorial constitutes the first part of a two- +part series on distributed optimization applied to multi-robot +problems, which seeks to advance the application of distributed +optimization in robotics. In this tutorial, we demonstrate that +many canonical multi-robot problems can be cast within the +distributed optimization framework, such as multi-robot simul- +taneous localization and planning (SLAM), multi-robot target +tracking, and multi-robot task assignment problems. We identify +three broad categories of distributed optimization algorithms: +distributed first-order methods, distributed sequential convex +programming, and the alternating direction method of multipliers +(ADMM). We describe the basic structure of each category +and provide representative algorithms within each category. +We then work through a simulation case study of multiple +drones collaboratively tracking a ground vehicle. We compare +solutions to this problem using a number of different distributed +optimization algorithms. In addition, we implement a distributed +optimization algorithm in hardware on a network of Rasberry +Pis communicating with XBee modules to illustrate robustness +to the challenges of real-world communication networks. +Index Terms—distributed optimization, multi-robot systems, +distributed robot systems, robotic sensor networks +I. INTRODUCTION +Distributed optimization is the problem of minimizing a joint +objective function subject to constraints using an algorithm +implemented on a network of communicating computation +nodes. In this tutorial, we specifically consider the computation +nodes as robots and the network as a typical multi-robot +mesh network. While distributed optimization has been a +longstanding topic of research in the optimization community +(e.g., [1], [2]), its usage in multi-robot systems is limited to only +a handful of examples. However, we note that many problems +in multi-robot coordination and collaboration can be formulated +and solved within the framework of distributed optimization, +including cooperative estimation [3], distributed SLAM, multi- +agent learning [4], and collaborative motion planning [5]. +This tutorial constitutes the first part of a two-part series +on distributed optimization methods for multi-robot systems. +While this tutorial focuses on the amenability of a broad +class of multi-robot problems to distributed optimization, +the second part of the series provides a survey of existing +*This project was funded in part by NSF NRI awards 1830402 and 1925030. +The first author was supported on an NDSEG Fellowship, and the third author +was supported on an NSF Graduate Research Fellowship. +1Department of Mechanical Engineering, Stanford University, Stanford, CA +94305, USA, {halsted, shorinwa}@stanford.edu +2Department of Aeronautics and Astronautics, Stanford University, Stanford, +CA 94305, USA {javieryu, schwager}@stanford.edu +distributed optimization methods, with an emphasis on their +applications to multi-robot problems, and highlights open +research problems in distributed optimization for multi-robot +systems. In this tutorial, we introduce the framework of +distributed optimization, noting the important components of +this framework. In distributed optimization, several robots +(agents/computational nodes) collectively optimize a shared +decision variable, where each agent has knowledge of only +its local objective and constraint function, without knowing +the objective and constraint functions of other agents. The +joint objective function is typically taken as the sum over all +the robots of their local objectives, and the constraints are +the union of their local constraints. As a result, each agent +only has partial knowledge of the joint optimization problem +and cannot solve the joint problem individually. We assume +that each robot can communicate with its immediate one-hop +neighbors over a robot-to-robot communication network, which +is representative of the communication network present in +many robotics problems. We do not include the many existing +algorithms that require multi-hop communication or require a +hub-spoke network topology, where all agents communicate to +a central hub, as these network requirements are incompatible +with the typical multi-robot network model. +Although the joint problem can be solved through central- +ized optimization, centralized optimization techniques require +significant computation and communication resources due to +the need for data aggregation at a central hub, making this +approach less appealing in a multi-robot setting, especially with +a large number of robots. Distributed optimization algorithms +enable each robot to solve for an optimal solution of the joint +optimization problem locally, in parallel, while communicating +with its one-hop neighbors over a communication network. In +general, these algorithms are iterative, with each robot sharing +its intermediate decision variables or problem gradients with +its neighbors at each iteration. However, robots do not share +information about their local objective and constraint functions +or their problem data, such as local measurements, leading to +an inherent privacy property for these algorithms. The decision +variables of all the robots converge to a common solution of +the optimization problem as the algorithm proceeds. In convex +problems, the solution computed by each robot corresponds to +a globally optimal solution; however, in non-convex problems, +the iterates of each robot generally converge to a locally optimal +solution. This tutorial is directed towards robotics practitioners +who want to derive the benefits of distributed optimization +in robotics, in addition to robotics researchers interested in +research encompassing distributed optimization and multi-robot +systems. +arXiv:2301.11313v1 [cs.RO] 26 Jan 2023 + +2 +We demonstrate in this tutorial that a number of major multi- +robot problems can be formulated as distributed optimization +problems, such as multi-robot simultaneous localization and +mapping (SLAM), multi-robot target tracking, multi-robot +task assignment, collaborative trajectory planning, and multi- +robot learning. We describe the general strategies useful in +reformulating robotics problems into a form amenable to +distributed optimization algorithms. We note that optimization- +based approaches often provide flexibility in solving many +robotics problems. For example, multi-robot target tracking +problems are typically solved via filtering or smoothing +approaches, posing certain challenges such as cross-correlation +of local measurements [6]. In contrast, formulating multi-robot +target tracking problems as optimization problems eliminates +these drawbacks. +We categorize distributed optimization algorithms into the +following classes: distributed first-order methods, distributed +sequential convex programming, and the alternating direction +method of multipliers (ADMM). We describe the defining math- +ematical technique of each category and provide representative +algorithms for each category. Lastly, we provide a case study +on multi-drone ground vehicle tracking, where we demonstrate +the application of distributed optimization to the target tracking +problem in simulation, and in addition, implement a distributed +optimization algorithm on hardware using the XBee DigiMesh +2.4 network modules for communication. +A. Contributions +This tutorial paper has three primary objectives: +1) Demonstrate the formulation of many canonical multi- +robot problems as distributed optimization problems. +2) Describe our three main classes of distributed optimiza- +tion algorithms, noting the distinct optimization technique +employed by algorithms within each category. +3) Provide a case study applying distributed optimization +to multi-robot problems in simulation, on a multi-drone +target tracking problem, and on hardware. +B. Organization +We present notation and mathematical preliminaries in +Section II and formulate the general distributed optimization +problem in Section III. In Section IV, we demonstrate that +many multi-robot problems can be cast within the framework +of distributed optimization. Section V describes our three main +categories of distributed optimization algorithms, noting the +distinguishing characteristics of each category. In addition, we +provide representative algorithms for each category in Section +V. Section VI gives a demonstration of distributed optimization +algorithms applied to a multi-drone vehicle tracking problem +in simulation and hardware. We offer concluding remarks in +Section VII. +II. NOTATION AND PRELIMINARIES +In this section, we introduce the notation used in this paper +and provide the definitions of mathematical concepts relevant +to the discussion of the distribution optimization algorithms. +We denote the gradient of a function f : Rn → R as ∇f and +its Hessian as ∇2f. We denote the vector containing all ones as +1n, where n represents the number of elements in the vector. +We discuss some relevant notions of the connectivity of a +graph. +Definition 1 (Connectivity of an Undirected Graph). An +undirected graph G is connected if a path exists between every +pair of vertices (i, j) where i, j ∈ V. Note that such a path +might traverse other vertices in G. +Definition 2 (Connectivity of a Directed Graph). A directed +graph G is strongly connected if a directed path exists between +every pair of vertices (i, j) where i, j ∈ V. In addition, a +directed graph G is weakly connected if the underlying +undirected graph is connected. The underlying undirected graph +Gu of a directed graph G refers to a graph with the same set of +vertices as G and a set of edges obtained by considering each +edge in G as a bi-directional edge. Consequently, every strongly +connected directed graph is weakly connected; however, the +converse is not true. +Definition 3 (Stochastic Matrix). A non-negative matrix +W ∈ Rn×n is referred to as a row-stochastic matrix if +W1n = 1n, +(1) +in other words, the sum of all elements in each row of the +matrix equals one. We refer to W as a column-stochastic matrix +if +1⊤ +n W = 1⊤ +n . +(2) +Likewise, for a doubly-stochastic matrix W, +W1n = 1n and 1⊤ +n W = 1⊤ +n . +(3) +In distributed optimization in multi-robot systems, robots +perform communication and computation steps to minimize +some joint objective function. We focus on problems in +which the robots’ exchange of information must respect the +topology of an underlying distributed communication graph, +which could possibly change over time. This communication +graph, denoted as G(t) = (V(t), E(t)), consists of vertices +V(t) = {1, . . . , N} and edges E(t) ⊆ V(t) × V(t) over which +pairwise communication can occur. For undirected graphs, we +denote the set of neighbors of robot i as Ni(t). For directed +graphs, we refer to the set of robots which can send information +to robot i as the set of in-neighbors of robot i, denoted by +N + +i (t). Likewise, for directed graphs, we refer to the set of +robots which can receive information from robot i as the out- +neighbors of robot i, denoted by N − +i (t). +III. PROBLEM FORMULATION +We consider a general distributed optimization problem +where each robot i ∈ V has access to its local objective +function fi : Rn → R but has no knowledge of the local +objective function of other robots. Such problems arise in +many robotics applications where the local objective functions +depend on data collected locally by each robot, often in the +form of measurements taken by sensors attached to the robot. +In addition to local knowledge of the local objective functions, + +3 +we assume that constraints on the optimization variable are only +known locally as well. The robots seek to collectively solve +a joint optimization problem, which consists of a separable +objective function, with each component known by only a +single robot, expressed as +min +x +� +i∈V +fi(x) +subject to gi(x) = 0 +∀i ∈ V +hi(x) ≤ 0 +∀i ∈ V +(4) +where x ∈ Rn denotes the joint optimization variable, gi(x) +denotes the equality constraint function of robot i, and hi(x) +denotes its inequality constraint function. We note that not all +robots need to have a local constraint function. In these cases, +the corresponding constraint functions are omitted in (4). +In many robotics problems, privacy concerns, as well as +the limited availability of computation and communication +resources, prevent each robot from sharing its local objective +and constraint functions with other robots. As such, we +focus on distributed algorithms that enable each robot to +compute the optimal solution of the joint problem in (4), +without sharing its local problem data and functions. In these +distributed algorithms, each robot maintains a local copy of +the optimization variable, with xi denoting robot i’s local +optimization variable. Distributed optimization algorithms solve +a reformulation of the optimization problem (4), given by +min +{xi, ∀i∈V} +� +i∈V +fi(xi) +subject to xi = xj +∀(i, j) ∈ E +gi(xi) = 0 +∀i ∈ V +hi(xi) ≤ 0 +∀i ∈ V. +(5) +We call the xi = xj +∀(i, j) ∈ E the consensus constraints. +Under the assumption that the communication graph is con- +nected for undirected graphs and weakly connected for directed +graphs, the optimal cost in (5) is equivalent to that in (4), and +the minimizing arguments x∗ +i in (5) are equal to the minimizing +argument x∗ of (4) for all robots i = 1, . . . , n. +IV. MULTI-ROBOT PROBLEMS POSED AS DISTRIBUTED +OPTIMIZATIONS +Many canonical robotics problems can be cast within the +framework of distributed optimization. In this section, we +consider five general problem categories that can be solved +using distributed optimization tools: multi-robot SLAM, multi- +robot target tracking, multi-robot task assignment, collabo- +rative planning, and multi-robot learning. We note that an +optimization-based approach to solving some of these problems +might not be immediately obvious. However, we show that +many of these problems can be quite easily formulated as +distributed optimization problems through the introduction of +auxiliary optimization variables, in addition to an appropriate +set of consensus constraints. +𝑥!,#$% +𝑥!,#$& +𝑥!,#$' +𝑥!,# +𝑧̂!,#$' +𝑧̆! +' +𝑚' +𝑚& +𝑚% +𝑧̆! +( +𝑚( +𝑚) +𝑧̂!,#$& +𝑧̂!,#$% +𝑧̆! +& +𝑧̆! +% +𝑧̆! +) +𝑥*,#$% +𝑥*,#$& +𝑥*,#$' +𝑥*,# +𝑧̂*,#$' +𝑧̂*,#$& +𝑧̂*,#$% +𝑧̆* +' +𝑧̆* +& +𝑧̆* +% +Fig. 1. A factor graph representation of a multi-robot SLAM problem, where +two robots, robot i (blue circles) and j (green circles), seek to jointly estimate +a set of map features {m1, m2, · · · } (orange triangles) in addition to their +own pose trajectory {xi,t, xj,t, ∀t}, from the set of odometry measurements +{ˆzi,t, ˆzj,t} and observations of each map feature k {˘zk +i , ˘zk +j }. +A. Multi-Robot Simultaneous Localization and Mapping +(SLAM) +In multi-robot simultaneous localization and mapping +(SLAM) problems, a group of robots seek to estimate their +position and orientation (pose) within a consistent represen- +tation of their environment. In a full landmark-based SLAM +approach, we consider optimizing over both map features as +well as robot poses: +minimize +x,m +N +� +i=1 +T −1 +� +t=0 +∥¯zi,t(xi,t, xi,t+1) − ˆzi,t+1∥2 +Ωi,t ++ +N +� +i=1 +M +� +k=1 +∥˜zk +i (xi, mk) − ˘zk +i ∥2 +Λi,t, +(6) +where there are N robots and M map features over a duration +of T + 1 timesteps, and the expected relative poses ¯zi,t are +functions of two adjacent poses of robot i derived from robot +odometry measurements, and the expected relative pose ˜zk +i +is a function of the pose of robot i and the position of map +feature k. We have concatenated the problem variables in (6), +with xi = +� +x⊤ +i,0, x⊤ +i,1, · · · , x⊤ +i,T +�⊤, x = +� +x⊤ +1 , x⊤ +2 , · · · , x⊤ +N +�⊤, +and m = +� +m⊤ +1 , m⊤ +2 , · · · , m⊤ +M +�⊤. The error terms in the ob- +jective function are weighted by the information matrices Ωi,t +and Λi,t associated with the measurements collected by robot +i. +Although the first set of terms in the objective function +of the optimization problem (6) is separable among the +robots, the second set of terms is not. Consequently, the +optimization problem must be reformulated for amenability +to distributed optimization algorithms. Non-separability of +the objective function arises from the coupling between the +map features and the robot poses. To achieve separability of +the objective function, we can introduce local copies of the +variables corresponding to each feature, with an associated set +of consensus (equality) constraints to ensure that the resulting +problem remains equivalent to the original problem (6). The + +4 +resulting problem takes the form +minimize +x, ˆm1, ˆm2,··· , ˆmN +N +� +i=1 +T −1 +� +t=0 +∥¯zi,t(xi,t, xi,t+1) − ˆzi,t+1∥2 +Ωi,t ++ +N +� +i=1 +M +� +k=1 +∥˜zk +i (xi, ˆmi,k) − ˘zk +i ∥2 +Λi,t +subject to ˆmi = ˆmj +∀(i, j) ∈ E, +(7) +where robot i maintains ˆmi, its local copy of the map m. The +problem (7) is separable among the robots; in other words, its +objective function can be expressed in the form +f(x, ˆm1, ˆm2, · · · , ˆmN) = +N +� +i=1 +fi(xi, ˆmi), +(8) +where +fi(xi, ˆmi) = +T −1 +� +t=0 +∥¯zi,t(xi,t, xi,t+1) − ˆzi,t+1∥2 +Ωi,t ++ +M +� +k=1 +∥˜zk +i (xi, ˆmi,k) − ˘zk +i ∥2 +Λi,t +. +(9) +We can interpret the bundle adjustment problem similarly— +in this case, the map features represent the scene geometry +and the robot poses include the optical characteristics of the +respective cameras. However, a challenge in applying this +approach in unstructured environments is ensuring that multiple +robots agree on the labels of the map landmarks. An alternative +approach is pose graph optimization, which avoids explicitly +estimating the map and instead uses relative pose measurements +based on shared observations of features in the environment. +In this perspective, multi-robot SLAM consists of a “front- +end,” in which the robots process raw sensor measurements +to generate relative pose measurements, and a “back-end,” in +which robots find optimal robot poses given those relative pose +measurements. The SLAM back-end is typically written as a +pose graph optimization problem, where edges between nodes +represent noisy relative pose estimates (which are obtained +from the front-end) derived from raw sensor measurements. +This pose graph optimization problem is a naturally separable +optimization which is amenable to distributed optimization +techniques, written as +min +{(Ri,τi)}n +i=1 +� +(i,j)∈E +ωij +2 ∥Rj −Ri ˜Rij∥2 +F + wij +2 ∥τj −τi −Ri˜τij∥2 +2 +The pose graph optimization problem seeks to minimize the +error between the expected relative pose obtained from the +estimated poses and the measured relative pose, summed over +all edges in the graph. The SLAM front end may be amenable to +distributed optimization techniques as well, although this is an +area of open research. Some existing distributed techniques that +do not rely on distributed optimization have been proposed +for the front-end, e.g., [7]. Refer to [8], [9], [10], [11] for +additional details on SLAM and multi-robot SLAM. +Hence, distributed optimization algorithms can be readily +applied to the graph-based SLAM problem in (7). More- +over, we note that a number of related robotics problems +— including rotation averaging/synchronization and shape +Fig. 2. A multi-robot target tracking scenario, with four quadrotors (the robots) +making noisy observations of the flagged ground vehicle (the target). The +colored cones represent the regions where each quadrotor can observe the +vehicle, given the limited measurement range of the sensors onboard each +quadrotor. +registration/alignment — can be similarly reformulated into +a separable form and subsequently solved using distributed +optimization algorithms [12], [13], [14], [15], [16], [17]. +B. Multi-Robot Target Tracking +In the multi-robot target tracking problem, a group of robots +collect measurements of an agent of interest (referred to as +a target) and seek to collectively estimate the trajectory of +the target. Multi-robot target tracking problems arise in many +robotics applications ranging from environmental monitoring +and surveillance to autonomous robotics applications such as +autonomous driving, where the estimated trajectory of the +target can be leveraged for scene prediction to enable safe +operation. Figure 2 provides an illustration of the multi-robot +target tracking problem where a group of four quadrotors make +noisy observations of the flagged ground vehicle (the target). +Each colored cone represents the region where each quadrotor +can observe the vehicle, given the limited measurement range +of the sensors onboard the quadrotor. +Multi-robot target tracking problems can be posed as +maximum a posterior (MAP) optimization problems where +the robots seek to compute an estimate that maximizes the +posterior distribution of the target’s trajectory given the set +of all observations of the target made by the robots. When +a model of the dynamics of target is available, denoted by +g : Rn → Rn, the resulting optimization problem takes the +form +minimize +x +T −1 +� +t=0 +∥xt+1 − g(xt)∥2 +Ωt ++ +N +� +i=1 +T −1 +� +t=0 +∥yi,t − hi(xt)∥2 +Λi,t, +(10) +where xt ∈ Rn denotes the pose of the target at time t and +yi,t ∈ Rm denotes robot i’s observation of the target at time t, +over a duration of T + 1 timesteps. We represent the trajectory +of the target with x = +� +x⊤ +0 , x⊤ +1 , · · · , x⊤ +T +�⊤. While the first term +in the objective function corresponds to the error between the +estimated state of the target at a subsequent timestep and its +expected state based on a model of its dynamics, the second +term corresponds to the error between the observations collected +by each robot and the expected measurement computed from the +estimated state of the target, where the function hi : Rn → Rm + +5 +denotes the measurement model of robot i. Further, the +information matrices Ωt ∈ Rn×n and Λi,t ∈ Rm×m for the +dynamics and measurement models, respectively, weight the +contribution of each term in the objective function appropriately, +reflecting prior confidence in the dynamics and measurement +models. The MAP optimization problem in (10) is not separable, +hence, not amenable to distributed optimization, in its current +form, due to coupling in the objective function arising from x. +Nonetheless, we can arrive at a separable optimization problem +through a fairly straightforward reformulation [3]. We can +assign a local copy of x to each robot, with ˆxi denoting robot +i’s local copy of x. The reformulated problem becomes +minimize +ˆx +N +� +i=1 +T −1 +� +t=0 +1 +N ∥ˆxi,t+1 − g(ˆxi,t)∥2 +Ωt ++ +N +� +i=1 +T −1 +� +t=0 +∥yi,t − hi(ˆxi,t)∥2 +Λi,t +subject to ˆxi = ˆxj +∀(i, j) ∈ E, +(11) +where ˆx = +� +ˆx⊤ +1 , ˆx⊤ +2 , · · · , ˆx⊤ +N +�⊤. Following this reformulation, +distributed optimization algorithms can be applied to compute +an estimate of the trajectory of the target from (11). +C. Multi-Robot Task Assignment +In the multi-robot task assignment problem, we seek an +optimal assignment of N robots to M tasks such that the total +cost incurred in completing the specified tasks is minimized. +However, we note that many task assignment problems consist +of an equal number of tasks and robots. The standard task +assignment problem has been studied extensively and is +typically solved using the Hungarian method [18]. However, +optimization-based methods have emerged as a competitive +approach due to their amenability to task assignment problems +with a diverse set of additional constraints, encoding individual +preferences or other relevant problem information, making +them a general-purpose approach. +The task assignment problem can be represented as a +weighted bipartite graph: a graph whose vertices can be +divide into two sets where no two nodes within a given +set share an edge. Further, each edge in the graph has an +associated weight. In task assignment problems, the edge +weight ci,j represents the cost of assigning robot i to task +j. Figure 3 depicts a task assignment problem represented +by a weighted bipartite graph, with three robots and three +tasks. Each robot knows its task preferences only and does +not know the task preferences of other robots. Equivalently, +the task assignment problem can be formulated as an integer +optimization problem. Many optimization-based methods solve +a relaxation of the integer optimization problem. Generally, in +problems with linear objective functions and affine constraints, +these optimization-based methods are guaranteed to yield an +optimal task assignment. The associated relaxed optimization +Tasks +Task 𝑗 +Robots +Robot 𝑖 +𝑐!,# +Fig. 3. +A multi-robot task assignment problem represented as a bipartite +graph, with three (Fetch) robots and three tasks. An edge with weight ci,j +between robot i and task j signifies the cost incurred by robot i if it performs +task j. In many problems, each robot’s task preferences (edge weights) is +neither known by other robots nor accessible to these robots. +problem takes the form +minimize +x +N +� +i=1 +c⊤ +i xi +subject to +N +� +i=1 +xi = 1M +1⊤ +Mxi = 1 +0 ≤ x ≤ 1, +(12) +where xi ∈ RM denotes the optimization variable of robot +i, representing its task assignment and x = [x1, x2, · · · , xN]. +Although the objective function of (12) is separable, the +optimization problem is not separable due to coupling of +the optimization variables arising in the first constraint. We +can obtain a separable problem, amenable to distributed +optimization, by assigning a local copy of x to each robot, +resulting in the problem +minimize +ˆx +N +� +i=1 +c⊤ +i ˆxi,i +subject to +N +� +i=1 +ˆxi,i = 1M +1⊤ +M ˆxi,i = 1 +0 ≤ ˆxi ≤ 1 +∀i ∈ V +ˆxi = ˆxj +∀(i, j) ∈ E +(13) +where ˆxi ∈ RM×N denotes robot i’s local copy of x and +ˆx = [ˆx0, ˆx1, · · · , ˆxN]. Although the reformulation in (13) is +simple, it does not scale efficiently with the number of robots +and tasks. A more efficient reformulation can be obtained +by considering the dual formulation of the task assignment +problem. For brevity, we omit a discussion of this approach +in this paper and refer readers to [19], [20], [21] where this +reformulation scheme is discussed in detail. +D. Collaborative Planning, Control, and Manipulation +Generally, in collaborative planning problems, we seek to +compute state and control input trajectories that enable a +group of robots to reach a desired state configuration from a +specified initial state, while minimizing a trajectory cost and + +freight100 +: +fetchrobatis6 +Fig. 4. A multi-robot manipulation problem, with three quadrotors collabora- +tively manipulating a load rigidly attached to each quadrotor. The dashed-line +represents the reference trajectory for manipulating the load. +without colliding with other agents. The related multi-robot +control problem involves computing a sequence of control +inputs that enables a group of robots to track a desired +reference trajectory or achieve some specified task such as +manipulating an object collaboratively. Figure 4 shows a +collaborative manipulation problem where three quadrotors +move an object collaboratively. The dashed-line represents the +reference trajectory for manipulating the load. +Collaborative multi-robot planning, control, and manipulation +problems have been well-studied, with a broad variety of +methods devised for these problems. Among these methods, +receding horizon or model predictive control (MPC) approaches +have received notable attention due to their flexibility in +encoding complex problem constraint and objectives. In MPC +approaches, these multi-robot problems are formulated as +optimization problems over a finite time duration at each +timestep. The resulting optimization problem is solved to obtain +a sequence of control inputs over the specified time duration; +however, only the initial control input is applied by each robot +at the current timestep. At the next timestep, a new optimization +problem is formulated, from which a new sequence of control +inputs is computed to obtain a new control input for that +timestep. This process is repeated until completion of the task. +At time t, the associated MPC optimization problem has the +form +minimize +x,u +N +� +i=1 +fi(x, u) +subject to g(x, u) = 0 +h(x, u) ≤ 0 +xi,0 = ¯xi +∀i ∈ V +(14) +where xi ∈ Rni denotes robot i’s state trajectory, ui ∈ Rmi de- +notes its control input trajectory, and x = +� +x⊤ +1 , x⊤ +2 , · · · , x⊤ +N +�⊤ +with u = +� +u⊤ +1 , u⊤ +2 , · · · , u⊤ +N +�⊤. The objective function of robot +i, fi : R¯n × R ¯m → R, is often quadratic, given by +fi(x, u) = (xi − ˜xi)⊤Qi(xi − ˜xi) ++ (ui − ˜ui)⊤Ri(ui − ˜ui), +(15) +where ˜xi and ˜ui denote the reference state and control input +trajectory, respectively, Qi ∈ Rni×ni and Ri ∈ Rmi×mi denote +the associated weight matrices for the terms in the objective +function, ¯n = �N +i=1 ni, and ¯m = �N +i=1 mi. The dynamics +function of the robots is encoded in g : R¯n × R ¯m → R¯n. Fur- +ther, other equality constraints can be encoded in g. Inequality +constraints, such as collision-avoidance constraints and other +state or control input feasibility constraints, are encoded in +h : R¯n × R ¯m → Rl. In addition, the first state variable of each +agent is constrained to be equal to its initial state, denoted +by ¯xi. In each instance of the MPC optimization problem, +the initial state ¯xi of robot i is specified as its current state +at that timestep. Note that the MPC optimization problem +in (14) is not generally separable, depending on the equality +and inequality constraints. However, a separable form of the +problem can always be obtained by introducing local copies +of the optimization variables that are coupled in (14). The +functions g and h can also encode complementarity constraints +for manipulation and locomotion problems that involve making +and breaking rigid body contact [22]. In the extreme case, +where the optimization variables are coupled in the objective +function and equality and inequality constraints in (14), a +suitable reformulation takes the form +minimize +ˆx,ˆu +N +� +i=1 +fi(ˆxi, ˆui) +subject to g(ˆxi, ˆui) = 0 +∀i ∈ V +h(ˆxi, ˆui) ≤ 0 +∀i ∈ V +φi(ˆxi) = ¯xi +∀i ∈ V +ˆxi = ˆxj +∀(i, j) ∈ E, +(16) +where the function φi outputs the first state variable corre- +sponding to robot i, given the input ˆxi, which denotes robot i’s +local copy of x. Similarly, ˆui denotes robot i’s local copy of +u, with ˆx = +� +ˆx⊤ +1 , ˆx⊤ +2 , · · · , ˆx⊤ +N +�⊤ and ˆu = +� +ˆu⊤ +1 , ˆu⊤ +2 , · · · , ˆu⊤ +N +�⊤. +Distributed optimization algorithms [5], [23], [24] can be +employed to solve the resulting MPC optimization problem in +(16). +E. Multi-Robot Learning +Multi-robot learning entails the application of deep learning +methods to approximate functions from data to solve multi- +robot tasks, such as object detection, visual place recognition, +monocular depth estimation, 3D mapping, and multi-robot rein- +forcement learning. Consider a general multi-robot supervised +learning problem where we aim to minimize a loss function +over labeled data collected by all the robots. We can write this +as +min +θ +N +� +i=1 +� +(xij,yij)∈Di +l(yi, f(xi; θ)), +where l(·, ·) is the loss function, (xij, yij) is data point +j collected by robot i with feature vector xij and label +yij, Di is the set of data collected by robot i, θ are the +neural network weights, and f(x; θ) is the neural network +parameterized function we desire to learn. By creating local +copies of the neural network weights θi and adding consensus +constraints θi = θj, we can put problem in the form (5), +so it is amenable to distributed optimization. We stress that +this problem encompasses a large majority of problems in + +7 +𝑎! +𝑜! +Observation 𝑜" +𝑜# +𝑎# +Action 𝑎" +Robot 𝑖 +Robot 𝑗 +Robot 𝑘 +Fig. 5. In multi-robot reinforcement learning problems, a group of robots +compute a control policy from experience by making sequential decisions while +interacting with their environment. Each robot takes an action and receives an +observation (and a reward), which provides information on its performance in +accomplishing a specified task. +supervised learning. See [25] for an ADMM-based distributed +optimization approach to solving this problem. +Beyond supervised learning, many multi-robot learning +problems are formulated within the framework of reinforcement +learning. In these problems, the robots learn a control policy +by interacting with their environments by making sequential +decisions. The underlying control policy, which drives these +sequential decisions, is iteratively updated to optimize the +performance of all agents on a specified objective using the +information gathered by each robot during its interaction with +its environment. Figure 5 illustrates the reinforcement learning +paradigm, where a group of robots learn from experience. +Each robot takes an action and receives an observation (and a +reward), which provides information on the performance of its +current control policy in achieving its specified objective. +Reinforcement learning approaches can be broadly cate- +gorized into value-based methods and policy-based methods. +Value-based methods seek to compute an estimate of the +optimal action-value function — the Q-function — which +represents the expected discounted reward starting from a +given state and taking a given action. An optimal policy can be +extracted from the estimated Q-function by selecting the action +that maximizes the value of the Q-function at a specified state. +In deep value-based methods, deep neural networks are utilized +in approximating the Q-function. In contrast, policy-based +methods seek to find an optimal policy by directly searching +over the space of policies. In deep policy-based methods, the +control policy is parameterized using deep neural networks. +In general, the agents seek to maximize the expected infinite- +horizon discounted cumulative reward, which is posed as the +optimization problem +maximize +θ +Eπθ +� +�� +t≥0 +γt +N +� +i=1 +Ri(si,t, ai,t) | si,0 = ¯si +� +� , +(17) +where πθ denotes the control policy parameterized by θ, γ ∈ R +denotes the discount factor (γ ∈ (0, 1)), si,t denotes the state of +robot i at time t, ai,t denotes its action at time t, ¯si denotes its +initial state, Ri : Si × Ai → R denotes the reward function of +robot i, and N denotes the number of robots. The optimization +problem in (17) is not separable in its current form. However, +due to the linearity of the expectation operator, the optimization +problem in (17) can be equivalently expressed as +maximize +ˆθ1,··· ,ˆθN +N +� +i=1 +Eπˆθi +� +�� +t≥0 +γtRi(si,t, ai,t) | si,0 = ¯si +� +� +subject to ˆθi = ˆθj +∀(i, j) ∈ E, +(18) +which is separable among the N robots. Hence, the resulting +problem can be readily solved using distributed optimization +algorithms for reinforcement learning problems, such as +distributed Q-learning and distributed actor-critic methods [26], +[27], [28]. +V. CLASSES OF DISTRIBUTED OPTIMIZATION ALGORITHMS +In this section, we categorize distributed optimization al- +gorithms into three broad classes — Distributed First-Order +Methods, Distributed Sequential Convex Programming, and +Alternating Direction Method of Multipliers — and provide a +brief overview of each category, by considering a representative +distributed algorithm within each category. In the subsequent +discussion, we consider the separable optimization problem +in (5). In the tutorial spirit of this paper, we first consider +problems without the local equality and inequality constraint +functions, gi and hi. In the second paper in this series, we +include these constraint functions, as we give a survey of more +sophisticated distributed optimization algorithms. +Before describing the specific algorithms that solve dis- +tributed optimization problems, we first consider the general +framework that all of these approaches share. Each algorithm +progresses over discrete iterations k = 0, 1, . . . until conver- +gence. In general, each iteration consists of a communication +step and a computation step. Besides assuming that each +robot has the sole capability of evaluating its local objective +function fi, we also distinguish between the “internal” variables +P(k) +i +that the robot computes at each iteration k and the +“communicated” variables Q(k) +i +that the robot communicates to +its neighbors. Each algorithm also involves parameters R(k) +i +, +which generally require coordination among all of the robots +but can typically be assigned before deployment of the system. +In distributed optimization, all the robots seek to collectively +minimize the joint objective function in (5) while achieving +consensus on a common set of minimizing optimization +variables. In this paper, we distinguish between two distinct +perspectives on how consensus between the robots is achieved. +Distributed first-order methods and distributed sequential +convex programming methods enforce the consensus (equality) +constraints in (5) implicitly, while the alternating direction +method of multipliers enforces these constraints explicitly. +A. Distributed First-Order Algorithms +Gradient decent methods have been widely applied to solve +broad classes of optimization problems. In general, these +methods only require the computation of the gradient (i.e., +the first derivative of the objective and constraint functions); + +8 +hence, these methods are also referred to as first-order methods. +When applied to (5), the updates to the optimization variable +take the form +x(k+1) = x(k) − α(k)∇f(x(k)) +(19) +where α(k) denotes a diminishing step-size and ∇f(x(k)) +denotes the gradient of the objective function, given by +∇f(x) = +� +i∈V +∇fi(x). +(20) +From (20), computation of ∇f(x) requires knowledge of +the objective function of all robots, which is unavailable to +any individual robot, and thus requires aggregation of this +information at a central node. +Distributed First-Order (DFO) algorithms provide alternative +update schemes that circumvent this underlying challenge by +enabling each robot to utilize only its local gradients, while +communicating with its neighbors to reach consensus on a +common solution. We can further categorize DFO methods into +two broad subclasses: Adapt-Then-Combine (ATC) methods +and Combine-Then-Adapt (CTA) methods, based on the relative +order of the communication and computation procedures. In +ATC methods, each robot updates its local optimization variable +using its gradient prior to combining its local variable with +that of its neighbors, with the update procedure given by +x(k+1) +i += +� +j∈Ni∪{i} +wij(x(k) +j +− α(k)y(k) +j +), +(21) +where x(k) +j +∈ Rn denotes the local variable of neighboring +robot j and y(k) +j +denotes j’s local gradient y(k) +j += ∇fj(x(k) +j ). +Note that each robot updates its local variable x(k+1) +i +using +gradients from its one-hop neighborhood before communicating +its local variable with its neighbors. The weight wi,j must +be compatible with the underlying communication network, +such that wi,j = 0 if robots i and j do not share a direct +communication link, and the weighting matrix W should be a +stochastic matrix. +In contrast, in CTA methods, each robot combines its local +variable with that of its neighbors prior to incorporating its +local gradient, yielding the update procedure +x(k+1) +i += +� +j∈Ni∪{i} +wijx(k) +j +− α(k)y(k) +i +. +(22) +In the simplest case, y(k) +i += ∇fi(x(k) +i +), or more generally, +y(k) +i += ∂fi(x(k) +i +) (where ∂fi denotes the subgradient of fi), +yielding the canonical distributed subgradient method [29]. This +choice of y(k) +i +necessitates the use of a diminishing step-size, +which is often given by α(k+1) = α(0) +√ +k . +More sophisticated gradient tracking methods, for example +DIGing [30], employ an estimate of the average gradient +computed through dynamic average consensus with +y(k+1) +i += +� +j∈Ni∪{i} +wijy(k) +j ++ +� +∇fi(x(k+1) +i +) − ∇fi(x(k) +i +) +� +, +(23) +which does not require a diminishing step-size. At initialization +of the algorithm, all the robots select a common step-size. +Algorithm 1: DIGing +Initialization: k ← 0, x(0) +i +∈ Rn, y(0) +i += ∇fi(x(0) +i ) +Internal variables: P(k) +i += ∅ +Communicated variables: Q(k) +i += +� +x(k) +i +, yk +i +� +Parameters: R(k) +i += (α, wi) +do in parallel ∀i ∈ V +Communicate Q(k) +i +to all j ∈ Ni +Receive Q(k) +j +from all j ∈ Ni +x(k+1) +i += +� +j∈Ni∪{i} +wijx(k) +j +− αy(k) +i +y(k+1) +i += +� +j∈Ni∪{i} +wijy(k) +j ++ ∇fi(x(k+1) +i +) − ∇fi(x(k) +i +) +k ← k + 1 +while stopping criterion is not satisfied +Further, robot i initializes its local variables with x(0) +i +∈ Rn +and y(0) +i += ∂fi(x(0) +i ). Algorithm 1 summarizes the update +procedures in the distributed gradient tracking method DIGing +[30]. Many DFO methods impose additional restrictions on the +weighting matrix. For example, DIGing requires a doubly- +stochastic weighting matrix for undirected communication +networks. +B. Distributed Sequential Convex Programming +Sequential convex programming entails solving an op- +timization problem by computing a sequence of iterates, +representing the solution of a series of approximations of the +original problem. Newton’s method is a prime example of a +sequential convex programming method. In Newton’s method, +and more generally, quasi-newton methods, we take a quadratic +approximation of the objective function at an operating point +x(k), resulting in +˜f(x) = f(x(k)) + ∇f(x(k))⊤(x − x(k)) ++ 1 +2(x − x(k))⊤H(x(k))(x − x(k)), +(24) +where H(·) denotes the Hessian of the objective function, ∇2f, +or its approximation. Subsequently, we compute a solution to +the quadratic program, given by +x(k+1) = x(k) − H +� +x(k)�−1∇f(˜x), +(25) +which requires centralized evaluation of the gradient and +Hessian of the objective function. Distributed Sequential +Programming enable each robot to compute a local estimate +of the gradient and Hessian of the objective function, and thus +allows for the local execution of the update procedures. We +consider the NEXT algorithm [31] to illustrate this class of +distributed optimization algorithms. We assume that each robot +uses a quadratic approximation of the optimization problem +as its convex surrogate model U(·). In NEXT, each robot +maintains an estimate of the average gradient of the objective +function, as well as an estimate of the gradient of the objective +function excluding its local component (e.g., � +j̸=i fj(xi) for + +9 +robot i). At a current iterate x(k) +i +, robot i creates a quadratic +approximation of the optimization problem, given by +minimize +˜xi +� +∇fi(x(k) +i +) + ˜π(k) +i +�⊤ � +˜xi − x(k) +i +� ++ 1 +2 +� +˜xi − x(k) +i +�⊤Hi +� +x(k) +i +�� +˜xi − x(k) +i +� +, +(26) +where ˜π(k) +i +denotes robot i’s estimate of the gradient of +� +j̸=i fj(xi) at x(k) +i +, which can be solved locally. Each robot +computes a weighted combination of its current iterate and the +solution of (26), given by the procedure +z(k) +i += x(k) +i ++ α(k) � +˜x(k) +i +− x(k) +i +� +, +(27) +where α(k) ∈ (0, 1) denotes a diminishing step-size. Subse- +quently, robot i computes its next iterate by taking a weighted +combination of its local estimate z(k) +i +with that of its neighbors +via the procedure +x(k+1) +i += +� +j∈Ni∪{i} +wijz(k) +j +, +(28) +for consensus on a common solution of the original optimiza- +tion problem, where the weight wi,j must be compatible with +the underlying communication network. In addition, robot i +updates its estimates of the average gradient of the objective +function, denoted by yi, using dynamic average consensus, +given by +y(k+1) +i += +� +j∈Ni∪{i} +wijy(k) +j ++ +� +∇fi(x(k+1) +i +) − ∇fi(x(k) +i +) +� +, +(29) +as well as ˜π(k) +i +using the procedure +˜π(k+1) +i += N · y(k+1) +i +− ∇fi(x(k+1) +i +). +(30) +Each agent initializes its local variables with x(0) +i +∈ Rn, +y(0) +i += ∇fi(x(0) +i ), and ˜π(k+1) +i += Ny(0) +i +− ∇fi(x(0) +i ), prior to +executing the above update procedures. Algorithm 2 summa- +rizes the update procedures in NEXT [31]. +C. Alternating Direction Method of Multipliers +The alternating direction method of multipliers (ADMM) +belongs to the class of optimization algorithms referred +to as the method of multipliers (or augmented Lagrangian +methods), which compute a primal-dual solution pair of a given +optimization problem. The method of multipliers proceeds +in an alternating fashion: the primal iterates are updated as +minimizers of the augmented Lagrangian, and subsequently, +the dual iterates are updated via dual (gradient) ascent on the +augmented Lagrangian. The procedure continues iteratively +until convergence or termination. The augmented Lagrangian +of the problem in (5) (with only the consensus constraints) is +given by +La(x, q) = +N +� +i=1 +fi(xi) ++ +N +� +i=1 +� +j∈Ni +� +q⊤ +i,j(xi − xj) + ρ +2∥xi − xj∥2 +2 +� +, +(31) +Algorithm 2: NEXT +Initialization: k ← 0, x(0) +i +∈ Rn, y(0) +i += ∇fi(x(0) +i ), +˜π(k+1) +i += Ny(0) +i +− ∇fi(x(0) +i ) +Internal variables: Pi = +� +x(k) +i +, ˜x(k) +i +, ˜π(k) +i +� +Communicated variables: Q(k) +i += +� +z(k) +i +, y(k) +i +� +Parameters: R(k) +i += +� +α(k), wi, U(·), K +� +do in parallel ∀i ∈ V +˜x(k) +i += argmin +x∈K +U +� +x; x(k) +i +, ˜π(k) +i +� +z(k) +i += x(k) +i ++ α(k) � +˜x(k) +i +− x(k) +i +� +Communicate Q(k) +i +to all j ∈ Ni +Receive Q(k) +j +from all j ∈ Ni +x(k+1) +i += +� +j∈Ni∪{i} +wijz(k) +j +y(k+1) +i += +� +j∈Ni∪{i} +wijy(k) +j ++ +� +∇fi(x(k+1) +i +) − ∇fi(x(k) +i +) +� +˜π(k+1) +i += N · y(k+1) +i +− ∇fi(x(k+1) +i +) +k ← k + 1 +while stopping criterion is not satisfied +where qi,j represents a dual variable for the consensus con- +straints between robots i and j, q = +� +q⊤ +i,j, ∀(i, j) ∈ E +�⊤, and +x = +� +x⊤ +1 , x⊤ +2 , · · · , x⊤ +N +�⊤. The parameter ρ > 0 represents a +penalty term on the violations of the consensus constraints. +Generally, the method of multipliers computes the minimizer +of the augmented Lagrangian with respect to the joint set of op- +timization variables, which hinders distributed computation. In +contrast, in the alternating direction method of multipliers, the +minimization procedure is performed block-component-wise, +enabling parallel, distributed computation of the minimization +subproblem in the consensus problem. However, many ADMM +algorithms still require some centralized computation, rendering +them not fully-distributed in multi-robot mesh network sense +that we consider in this paper. +We focus here on ADMM algorithms that are distributed over +robots in a mesh network, with each robot executing the same +set of distributed steps. We specifically consider the consensus +alternating direction method of multipliers (C-ADMM) [32] +as a representative algorithm within this category. C-ADMM +introduces auxiliary optimization variables into the consensus +constraints in (5) to enable fully-distributed update procedures. +The primal update procedure of robot i takes the form +x(k+1) +i += argmin +xi +� +fi(xi) + x⊤ +i y(k) +i ++ ρ +� +j∈Ni +����xi − 1 +2 +� +x(k) +i ++ x(k) +j +����� +2 +2 +� +, +(32) + +10 +Algorithm 3: C-ADMM +Initialization: k ← 0, x(0) +i +∈ Rn, y(0) +i += 0 +Internal variables: P(k) +i += y(k) +i +Communicated variables: Q(k) +i += x(k) +i +Parameters: R(k) +i += ρ +do in parallel ∀i ∈ V +x(k+1) +i += argmin +xi +� +fi(xi) + x⊤ +i y(k) +i +· · · ++ ρ +� +j∈Ni +����xi − 1 +2 +� +x(k) +i ++ x(k) +j +����� +2 +2 +� +Communicate Q(k) +i +to all j ∈ Ni +Receive Q(k) +j +from all j ∈ Ni +y(k+1) +i += y(k) +i ++ ρ +� +j∈Ni +� +x(k+1) +i +− x(k+1) +j +� +k ← k + 1 +while stopping criterion is not satisfied +which only requires information locally available to robot +i, including information received from its neighbors (i.e., +xk +j , ∀j ∈ Ni). As a result, this procedure can be executed +locally by each agent, in parallel. After communicating with +its neighbors, each robot updates its local dual variable using +the procedure +y(k+1) +i += y(k) +i ++ ρ +� +j∈Ni +� +x(k+1) +i +− x(k+1) +j +� +, +(33) +where yi denotes the composite dual variable of robot i, +corresponding to the consensus constraints between robot i +and its neighbors, which is initialized to zero. Algorithm 3 +summarizes the update procedures in C-ADMM [32]. +VI. DISTRIBUTED MULTI-DRONE VEHICLE TRACKING: A +CASE STUDY +Many robotics problems have a distributed structure, al- +though this structure might not be immediately apparent. In +many cases, applying distributed optimization methods requires +reformulating the original problem into a separable form that +allows for distributed computation of the problem variables +locally by each robot. This reformulation often involves the +introduction of additional problem variables local to each robot +with an associated set of constraints relating the local variables +between the robots. We illustrate this procedure using multi- +drone vehicle target tracking as a case study in simulation. We +note that the same principles apply to a broad class of robotics +problems as we have outlined in Sec. IV. In addition, we +implement the distributed optimization algorithm C-ADMM +on hardware, to demonstrate the deployment of distributed +optimization algorithms on hardware. +A. Simulation Study +In this simulation, we consider a distributed multi-drone +vehicle target tracking problem in which robots connected +by a communication graph, G = (V, E), each record range- +limited linear measurements of a moving target, and seek to +collectively estimate the target’s entire trajectory. We assume +that each drone can communicate locally with nearby drones +over the undirected communication graph G. The drones all +share a linear model of the target’s dynamics as +xt+1 = Atxt + wt, +(34) +where xt ∈ R4 represents the position and velocity of the +target in some global frame at time t, At is the dynamics +matrix associated with a linear model of the target’s dynamics, +and wt ∼ N(0, Qt) represents process noise (including the +unknown control inputs to the target). Restricting our case +study to a linear target model in this tutorial ensures that the +underlying optimization problem is convex, leading to strong +convergence guarantees and robust numerical properties for our +algorithm. A more expressive nonlinear model can also be used, +but this requires a more sophisticated distributed optimization +algorithm with more challenging numerical properties. At every +time-step when the target is sufficiently close to a drone i +(which we denote by t ∈ Ti), that robot collects an observation +according to the linear measurement model +yi,t = Ci,txt + vi,t , +(35) +where yi,t ∈ R2 is a positional measurement, Ci,t is the +measurement matrix of drone i, and vi,t ∼ N(0, Ri,t) is +measurement noise. We again assume a linear measurement +model to keep this case study as simple as possible. A nonlinear +model can also be used. +All of the drones have the same model for the prior distribu- +tion of the initial state of the target N(¯x0, ¯P0), where ¯x0 ∈ R4 +denotes the mean and ¯P0 ∈ R4×4 denotes the covariance. The +global cost function is of the form +f(x) =∥x0 − ¯x0∥2 +¯ +P −1 +0 ++ +T −1 +� +t=1 +∥xt+1 − Atxt∥2 +Q−1 +t ++ +� +i∈V +� +t∈Ti +∥yi,t − Ci,txt∥2 +R−1, +(36) +while the local cost function for drone i is +fi(x) = 1 +N ∥x0 − ¯x0∥2 +¯ +P −1 +0 ++ +T −1 +� +t=1 +1 +N ∥xt+1 − Atxt∥2 +Q−1 +t ++ +� +t∈Ti +∥yi,t − Ci,txt∥2 +R−1. +(37) +In our results, we consider only a batch solution to the +problem (finding the full trajectory of the target given each +robot’s full set of measurements). Methods for performing the +estimate in real-time through filtering and smoothing steps have +been well studied, both in the centralized and distributed case +[33]. An extended version of this multi-robot tracking problem +is solved with distributed optimization in [3]. A rendering of +a representative instance of this multi-robot tracking problem +is shown in Figure 2. +In Figures 6 and 7, several distributed optimization algo- +rithms are compared on an instance of the distributed multi- +drone vehicle tracking problem. For this problem instance, 10 + +11 +Fig. 6. Mean Square Error (MSE) per iteration on a distributed multi-drone +vehicle target tracking problem with N = 10 and n = 64. +simulated drones seek to estimate the target’s trajectory over +16 time steps resulting in a decision variable dimension of +n = 64. We compare four distributed optimization methods +which we consider to be representative of the taxonomic +classes outlined in the sections above: C-ADMM [32], EXTRA +[34], DIGing [30], and NEXT-Q [31]. Figure 6 shows that +C-ADMM and EXTRA have similar fast convergence rates +per iteration while DIGing and NEXT-Q are 4 and 15 times +slower respectively to converge below an MSE of 10−6. The +step-size hyperparameters for each method are computed by +Golden Section Search (GSS) (for NEXT-Q, which uses a two +parameter decreasing step-size, we fix one according to the +values recommended in [31]). +We note that tuning is essential for achieving robust +and efficient convergence with most distributed optimization +algorithms. Figure 7 shows the sensitivity of these methods to +variation in step-size, and highlights that three of the methods +(all except C-ADMM) become divergent for certain subsets +of the tested hyperparameter space. While C-ADMM seems +to be the most effective algorithm in this problem instance, +we note that other algorithms may have properties that are +advantageous in other instances of this problem or other +problems. For example, C-ADMM is known to require tight +synchronization among the robots (or the computing nodes, +more generally). If a robot misses a message or if robot clocks +are not precisely synchronized, C-ADMM can perform poorly. +First order methods, such as DIGing, tend to be more robust to +asynchronicity, for example, and modifying these algorithms +totolerate real-world challenges such as asynchronicity remains +an area of active research. We discuss such issues in the second +paper in this series. +B. Hardware Implementation +In this section, we discuss our implementation of the +C-ADMM algorithm on hardware. Each robot is equipped with +local computational resources and communication hardware +Fig. 7. Step-size hyperparameter sensitivity sweep on a distributed multi-drone +vehicle target tracking problem for N = 10 and n = 64. The dashed lines +are thresholds for divergence (top) and convergence (bottom) in terms of MSE +after 104 decision variable updates. +necessary for peer-to-peer communication with other neighbor- +ing robots. In the following discussion, we provide details of +the hardware platform, the underlying communication network +between robots, and the optimization problem considered in +this section. +We consider the linear least-squares optimization problem +min +p +N +� +i=1 +(Gip − zi)⊤Mi(Gip − zi), +(38) +with +the +optimization +variable +p ∈ R32, +Gi ∈ Rmi×32, +Mi ∈ Rmi×mi, zi ∈ Rmi, and N = 3 robots, where mi de- +pends on the number of measurements available to robot +i. In this experiment, we have m1 = 3268, m2 = 5422, and +m3 = 3528. We implement C-ADMM to solve the problem, +with a state size consisting of 32 floating-point variables. +The core communication infrastructure that we use are Digi +XBee DigiMesh 2.4 radio frequency mesh networking modules +which allow for peer-to-peer communication between robots. +Local computation for each robot is performed using Raspberry +Pi 4B single board computers. The lower level mesh network +is managed by the DigiMesh software, and we interact with it +through XBee Python Library. +We utilize the neighbor discovery Application Programming +Interface (API) provided by Digi International to enable each +robot to identify other neighboring robots. This approach +resulted in a fully-connected communication network, con- +sidering the XBee radios have an indoor range of up to 90m +and an outdoor range of up to 1500m. The XBee modules +used in our experiments have a maximum payload size of +92 bytes. However, the local variable of each robot in our +experiment consists of 32 floating-point variables, which +exceeds the maximum payload size that can be transmitted +by the XBee radios at each broadcast round, presenting a +communication challenge. To overcome this challenge, we +break up the local variables into a series of packets of size + +12 +0 +50 +100 +150 +200 +250 +Number of Iterations +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +‖x − x⋆‖2 +2 +Fig. 8. Convergence of the iterates computed by each robot using C-ADMM, +implemented on hardware, on the optimization problem with three robots in +(38). The convergence errors of all the robots overlap in the figure. +92 bytes and perform multiple broadcast rounds. We note, +however, that other approaches could be employed to overcome +the limited communication bandwidth of the XBee radios, +including utilizing quantization-based distributed optimization +methods. +We set the penalty parameter in C-ADMM to a value of 5 +and do not perform a comprehensive search for the penalty +parameter. In our experiments, we noticed that this value of +the penalty parameter provided suitable performance. Noting +that C-ADMM requires synchronous updates, we ensure that +the local clocks of all robots remain synchronized in our +experiments using a barrier strategy, which prevents each robot +from advancing to the next iteration of C-ADMM until all +other robots have completed the current iteration. In Figure 8, +we show the convergence error between the iterates of each +robot and the global solution, which is obtained by aggregating +the local data of all robots and then computing the solution +centrally. The convergence errors of all the robots iterates +overlap in the figure, with the error decreasing below 10−5 +within 250 iterations, showing convergence of the local iterates +of each robot to the optimal solution. +VII. CONCLUSION +In this tutorial, we have demonstrated that a number of +canonical problems in multi-robot systems can be formulated +and solved through the framework of distributed optimization. +We have identified three broad classes of distributed optimiza- +tion algorithms: distributed first-order methods, distributed +sequential convex programming methods, and the alternat- +ing direction method of multipliers (ADMM). Further, we +have described the optimization techniques employed by the +algorithms within each category, providing a representative +algorithm for each category. In addition, we have demonstrated +the application of distributed optimization in simulation, on +a distributed multi-drone vehicle tracking problems, and on +hardware, showing the effectiveness of distributed optimization +algorithms. However, important challenges remain in develop- +ing distributed algorithms for constrained, non-convex robotics +problems, and algorithms tailored to the limited computation +and communication resources of robot platforms, which we +discuss in greater detail in the second paper in this series [35]. +ACKNOWLEDGMENT +The authors would like to thank Siddharth Tanwar for his +help in performing the hardware experiments. +REFERENCES +[1] R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” +SIAM journal on control and optimization, vol. 14, no. 5, pp. 877–898, +1976. +[2] J. N. Tsitsiklis, “Problems in decentralized decision making and +computation.” Massachusetts Inst of Tech Cambridge Lab for Information +and Decision Systems, Tech. Rep., 1984. +[3] O. Shorinwa, J. Yu, T. Halsted, A. Koufos, and M. 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Schwager, “Distributed Opti- +mization Methods for Multi-Robot Systems: Part II — A Survey,” 2023. + diff --git a/cdFIT4oBgHgl3EQfnSu4/content/tmp_files/load_file.txt b/cdFIT4oBgHgl3EQfnSu4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23508e24be916a4b90b3a5f2dc04b9da2ec95990 --- /dev/null +++ b/cdFIT4oBgHgl3EQfnSu4/content/tmp_files/load_file.txt @@ -0,0 +1,760 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf,len=759 +page_content='1 Distributed Optimization Methods for Multi-Robot Systems: Part I — A Tutorial Ola Shorinwa,1 Trevor Halsted,1 Javier Yu,2 Mac Schwager2 Abstract—Distributed optimization provides a framework for deriving distributed algorithms for a variety of multi-robot problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This tutorial constitutes the first part of a two- part series on distributed optimization applied to multi-robot problems, which seeks to advance the application of distributed optimization in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this tutorial, we demonstrate that many canonical multi-robot problems can be cast within the distributed optimization framework, such as multi-robot simul- taneous localization and planning (SLAM), multi-robot target tracking, and multi-robot task assignment problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We identify three broad categories of distributed optimization algorithms: distributed first-order methods, distributed sequential convex programming, and the alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We describe the basic structure of each category and provide representative algorithms within each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We then work through a simulation case study of multiple drones collaboratively tracking a ground vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We compare solutions to this problem using a number of different distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, we implement a distributed optimization algorithm in hardware on a network of Rasberry Pis communicating with XBee modules to illustrate robustness to the challenges of real-world communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Index Terms—distributed optimization, multi-robot systems, distributed robot systems, robotic sensor networks I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' INTRODUCTION Distributed optimization is the problem of minimizing a joint objective function subject to constraints using an algorithm implemented on a network of communicating computation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this tutorial, we specifically consider the computation nodes as robots and the network as a typical multi-robot mesh network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' While distributed optimization has been a longstanding topic of research in the optimization community (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=', [1], [2]), its usage in multi-robot systems is limited to only a handful of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, we note that many problems in multi-robot coordination and collaboration can be formulated and solved within the framework of distributed optimization, including cooperative estimation [3], distributed SLAM, multi- agent learning [4], and collaborative motion planning [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This tutorial constitutes the first part of a two-part series on distributed optimization methods for multi-robot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' While this tutorial focuses on the amenability of a broad class of multi-robot problems to distributed optimization, the second part of the series provides a survey of existing This project was funded in part by NSF NRI awards 1830402 and 1925030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The first author was supported on an NDSEG Fellowship, and the third author was supported on an NSF Graduate Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 1Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA, {halsted, shorinwa}@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='edu 2Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA {javieryu, schwager}@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='edu distributed optimization methods, with an emphasis on their applications to multi-robot problems, and highlights open research problems in distributed optimization for multi-robot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this tutorial, we introduce the framework of distributed optimization, noting the important components of this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In distributed optimization, several robots (agents/computational nodes) collectively optimize a shared decision variable, where each agent has knowledge of only its local objective and constraint function, without knowing the objective and constraint functions of other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The joint objective function is typically taken as the sum over all the robots of their local objectives, and the constraints are the union of their local constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' As a result, each agent only has partial knowledge of the joint optimization problem and cannot solve the joint problem individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We assume that each robot can communicate with its immediate one-hop neighbors over a robot-to-robot communication network, which is representative of the communication network present in many robotics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We do not include the many existing algorithms that require multi-hop communication or require a hub-spoke network topology, where all agents communicate to a central hub, as these network requirements are incompatible with the typical multi-robot network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Although the joint problem can be solved through central- ized optimization, centralized optimization techniques require significant computation and communication resources due to the need for data aggregation at a central hub, making this approach less appealing in a multi-robot setting, especially with a large number of robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed optimization algorithms enable each robot to solve for an optimal solution of the joint optimization problem locally, in parallel, while communicating with its one-hop neighbors over a communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In general, these algorithms are iterative, with each robot sharing its intermediate decision variables or problem gradients with its neighbors at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, robots do not share information about their local objective and constraint functions or their problem data, such as local measurements, leading to an inherent privacy property for these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The decision variables of all the robots converge to a common solution of the optimization problem as the algorithm proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In convex problems, the solution computed by each robot corresponds to a globally optimal solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' however, in non-convex problems, the iterates of each robot generally converge to a locally optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This tutorial is directed towards robotics practitioners who want to derive the benefits of distributed optimization in robotics, in addition to robotics researchers interested in research encompassing distributed optimization and multi-robot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='11313v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='RO] 26 Jan 2023 2 We demonstrate in this tutorial that a number of major multi- robot problems can be formulated as distributed optimization problems, such as multi-robot simultaneous localization and mapping (SLAM), multi-robot target tracking, multi-robot task assignment, collaborative trajectory planning, and multi- robot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We describe the general strategies useful in reformulating robotics problems into a form amenable to distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We note that optimization- based approaches often provide flexibility in solving many robotics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For example, multi-robot target tracking problems are typically solved via filtering or smoothing approaches, posing certain challenges such as cross-correlation of local measurements [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In contrast, formulating multi-robot target tracking problems as optimization problems eliminates these drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We categorize distributed optimization algorithms into the following classes: distributed first-order methods, distributed sequential convex programming, and the alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We describe the defining math- ematical technique of each category and provide representative algorithms for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Lastly, we provide a case study on multi-drone ground vehicle tracking, where we demonstrate the application of distributed optimization to the target tracking problem in simulation, and in addition, implement a distributed optimization algorithm on hardware using the XBee DigiMesh 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='4 network modules for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Contributions This tutorial paper has three primary objectives: 1) Demonstrate the formulation of many canonical multi- robot problems as distributed optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 2) Describe our three main classes of distributed optimiza- tion algorithms, noting the distinct optimization technique employed by algorithms within each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 3) Provide a case study applying distributed optimization to multi-robot problems in simulation, on a multi-drone target tracking problem, and on hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Organization We present notation and mathematical preliminaries in Section II and formulate the general distributed optimization problem in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In Section IV, we demonstrate that many multi-robot problems can be cast within the framework of distributed optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Section V describes our three main categories of distributed optimization algorithms, noting the distinguishing characteristics of each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, we provide representative algorithms for each category in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Section VI gives a demonstration of distributed optimization algorithms applied to a multi-drone vehicle tracking problem in simulation and hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We offer concluding remarks in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' NOTATION AND PRELIMINARIES In this section, we introduce the notation used in this paper and provide the definitions of mathematical concepts relevant to the discussion of the distribution optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We denote the gradient of a function f : Rn → R as ∇f and its Hessian as ∇2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We denote the vector containing all ones as 1n, where n represents the number of elements in the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We discuss some relevant notions of the connectivity of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Definition 1 (Connectivity of an Undirected Graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' An undirected graph G is connected if a path exists between every pair of vertices (i, j) where i, j ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Note that such a path might traverse other vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Definition 2 (Connectivity of a Directed Graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A directed graph G is strongly connected if a directed path exists between every pair of vertices (i, j) where i, j ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, a directed graph G is weakly connected if the underlying undirected graph is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The underlying undirected graph Gu of a directed graph G refers to a graph with the same set of vertices as G and a set of edges obtained by considering each edge in G as a bi-directional edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Consequently, every strongly connected directed graph is weakly connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' however, the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Definition 3 (Stochastic Matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A non-negative matrix W ∈ Rn×n is referred to as a row-stochastic matrix if W1n = 1n, (1) in other words, the sum of all elements in each row of the matrix equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We refer to W as a column-stochastic matrix if 1⊤ n W = 1⊤ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (2) Likewise, for a doubly-stochastic matrix W, W1n = 1n and 1⊤ n W = 1⊤ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (3) In distributed optimization in multi-robot systems, robots perform communication and computation steps to minimize some joint objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We focus on problems in which the robots’ exchange of information must respect the topology of an underlying distributed communication graph, which could possibly change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This communication graph, denoted as G(t) = (V(t), E(t)), consists of vertices V(t) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' , N} and edges E(t) ⊆ V(t) × V(t) over which pairwise communication can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For undirected graphs, we denote the set of neighbors of robot i as Ni(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For directed graphs, we refer to the set of robots which can send information to robot i as the set of in-neighbors of robot i, denoted by N + i (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Likewise, for directed graphs, we refer to the set of robots which can receive information from robot i as the out- neighbors of robot i, denoted by N − i (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' PROBLEM FORMULATION We consider a general distributed optimization problem where each robot i ∈ V has access to its local objective function fi : Rn → R but has no knowledge of the local objective function of other robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Such problems arise in many robotics applications where the local objective functions depend on data collected locally by each robot, often in the form of measurements taken by sensors attached to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition to local knowledge of the local objective functions, 3 we assume that constraints on the optimization variable are only known locally as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The robots seek to collectively solve a joint optimization problem, which consists of a separable objective function, with each component known by only a single robot, expressed as min x � i∈V fi(x) subject to gi(x) = 0 ∀i ∈ V hi(x) ≤ 0 ∀i ∈ V (4) where x ∈ Rn denotes the joint optimization variable, gi(x) denotes the equality constraint function of robot i, and hi(x) denotes its inequality constraint function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We note that not all robots need to have a local constraint function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In these cases, the corresponding constraint functions are omitted in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In many robotics problems, privacy concerns, as well as the limited availability of computation and communication resources, prevent each robot from sharing its local objective and constraint functions with other robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' As such, we focus on distributed algorithms that enable each robot to compute the optimal solution of the joint problem in (4), without sharing its local problem data and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In these distributed algorithms, each robot maintains a local copy of the optimization variable, with xi denoting robot i’s local optimization variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed optimization algorithms solve a reformulation of the optimization problem (4), given by min {xi, ∀i∈V} � i∈V fi(xi) subject to xi = xj ∀(i, j) ∈ E gi(xi) = 0 ∀i ∈ V hi(xi) ≤ 0 ∀i ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (5) We call the xi = xj ∀(i, j) ∈ E the consensus constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Under the assumption that the communication graph is con- nected for undirected graphs and weakly connected for directed graphs, the optimal cost in (5) is equivalent to that in (4), and the minimizing arguments x∗ i in (5) are equal to the minimizing argument x∗ of (4) for all robots i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' MULTI-ROBOT PROBLEMS POSED AS DISTRIBUTED OPTIMIZATIONS Many canonical robotics problems can be cast within the framework of distributed optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this section, we consider five general problem categories that can be solved using distributed optimization tools: multi-robot SLAM, multi- robot target tracking, multi-robot task assignment, collabo- rative planning, and multi-robot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We note that an optimization-based approach to solving some of these problems might not be immediately obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, we show that many of these problems can be quite easily formulated as distributed optimization problems through the introduction of auxiliary optimization variables, in addition to an appropriate set of consensus constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=',#$% 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=',#$& 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=",#$' 𝑥!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=',# 𝑧̂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=",#$' 𝑧̆!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=" ' 𝑚' 𝑚& 𝑚% 𝑧̆!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ( 𝑚( 𝑚) 𝑧̂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=',#$& 𝑧̂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=',#$% 𝑧̆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' & 𝑧̆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' % 𝑧̆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=" ) 𝑥*,#$% 𝑥*,#$& 𝑥*,#$' 𝑥*,# 𝑧̂*,#$' 𝑧̂*,#$& 𝑧̂*,#$% 𝑧̆* ' 𝑧̆* & 𝑧̆* % Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A factor graph representation of a multi-robot SLAM problem, where two robots, robot i (blue circles) and j (green circles), seek to jointly estimate a set of map features {m1, m2, · · · } (orange triangles) in addition to their own pose trajectory {xi,t, xj,t, ∀t}, from the set of odometry measurements {ˆzi,t, ˆzj,t} and observations of each map feature k {˘zk i , ˘zk j }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Multi-Robot Simultaneous Localization and Mapping (SLAM) In multi-robot simultaneous localization and mapping (SLAM) problems, a group of robots seek to estimate their position and orientation (pose) within a consistent represen- tation of their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In a full landmark-based SLAM approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' we consider optimizing over both map features as well as robot poses: minimize x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='m N � i=1 T −1 � t=0 ∥¯zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t+1) − ˆzi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t+1∥2 Ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t + N � i=1 M � k=1 ∥˜zk i (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' mk) − ˘zk i ∥2 Λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (6) where there are N robots and M map features over a duration of T + 1 timesteps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' and the expected relative poses ¯zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t are functions of two adjacent poses of robot i derived from robot odometry measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' and the expected relative pose ˜zk i is a function of the pose of robot i and the position of map feature k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We have concatenated the problem variables in (6), with xi = � x⊤ i,0, x⊤ i,1, · · · , x⊤ i,T �⊤, x = � x⊤ 1 , x⊤ 2 , · · · , x⊤ N �⊤, and m = � m⊤ 1 , m⊤ 2 , · · · , m⊤ M �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The error terms in the ob- jective function are weighted by the information matrices Ωi,t and Λi,t associated with the measurements collected by robot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Although the first set of terms in the objective function of the optimization problem (6) is separable among the robots, the second set of terms is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Consequently, the optimization problem must be reformulated for amenability to distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Non-separability of the objective function arises from the coupling between the map features and the robot poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' To achieve separability of the objective function, we can introduce local copies of the variables corresponding to each feature, with an associated set of consensus (equality) constraints to ensure that the resulting problem remains equivalent to the original problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The 4 resulting problem takes the form minimize x, ˆm1, ˆm2,··· , ˆmN N � i=1 T −1 � t=0 ∥¯zi,t(xi,t, xi,t+1) − ˆzi,t+1∥2 Ωi,t + N � i=1 M � k=1 ∥˜zk i (xi, ˆmi,k) − ˘zk i ∥2 Λi,t subject to ˆmi = ˆmj ∀(i, j) ∈ E, (7) where robot i maintains ˆmi, its local copy of the map m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The problem (7) is separable among the robots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' in other words, its objective function can be expressed in the form f(x, ˆm1, ˆm2, · · · , ˆmN) = N � i=1 fi(xi, ˆmi), (8) where fi(xi, ˆmi) = T −1 � t=0 ∥¯zi,t(xi,t, xi,t+1) − ˆzi,t+1∥2 Ωi,t + M � k=1 ∥˜zk i (xi, ˆmi,k) − ˘zk i ∥2 Λi,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (9) We can interpret the bundle adjustment problem similarly— in this case, the map features represent the scene geometry and the robot poses include the optical characteristics of the respective cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, a challenge in applying this approach in unstructured environments is ensuring that multiple robots agree on the labels of the map landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' An alternative approach is pose graph optimization, which avoids explicitly estimating the map and instead uses relative pose measurements based on shared observations of features in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this perspective, multi-robot SLAM consists of a “front- end,” in which the robots process raw sensor measurements to generate relative pose measurements, and a “back-end,” in which robots find optimal robot poses given those relative pose measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The SLAM back-end is typically written as a pose graph optimization problem, where edges between nodes represent noisy relative pose estimates (which are obtained from the front-end) derived from raw sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This pose graph optimization problem is a naturally separable optimization which is amenable to distributed optimization techniques, written as min {(Ri,τi)}n i=1 � (i,j)∈E ωij 2 ∥Rj −Ri ˜Rij∥2 F + wij 2 ∥τj −τi −Ri˜τij∥2 2 The pose graph optimization problem seeks to minimize the error between the expected relative pose obtained from the estimated poses and the measured relative pose, summed over all edges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The SLAM front end may be amenable to distributed optimization techniques as well, although this is an area of open research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Some existing distributed techniques that do not rely on distributed optimization have been proposed for the front-end, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=', [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Refer to [8], [9], [10], [11] for additional details on SLAM and multi-robot SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Hence, distributed optimization algorithms can be readily applied to the graph-based SLAM problem in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' More- over, we note that a number of related robotics problems — including rotation averaging/synchronization and shape Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A multi-robot target tracking scenario, with four quadrotors (the robots) making noisy observations of the flagged ground vehicle (the target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The colored cones represent the regions where each quadrotor can observe the vehicle, given the limited measurement range of the sensors onboard each quadrotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' registration/alignment — can be similarly reformulated into a separable form and subsequently solved using distributed optimization algorithms [12], [13], [14], [15], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Multi-Robot Target Tracking In the multi-robot target tracking problem, a group of robots collect measurements of an agent of interest (referred to as a target) and seek to collectively estimate the trajectory of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Multi-robot target tracking problems arise in many robotics applications ranging from environmental monitoring and surveillance to autonomous robotics applications such as autonomous driving, where the estimated trajectory of the target can be leveraged for scene prediction to enable safe operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Figure 2 provides an illustration of the multi-robot target tracking problem where a group of four quadrotors make noisy observations of the flagged ground vehicle (the target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each colored cone represents the region where each quadrotor can observe the vehicle, given the limited measurement range of the sensors onboard the quadrotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Multi-robot target tracking problems can be posed as maximum a posterior (MAP) optimization problems where the robots seek to compute an estimate that maximizes the posterior distribution of the target’s trajectory given the set of all observations of the target made by the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' When a model of the dynamics of target is available, denoted by g : Rn → Rn, the resulting optimization problem takes the form minimize x T −1 � t=0 ∥xt+1 − g(xt)∥2 Ωt + N � i=1 T −1 � t=0 ∥yi,t − hi(xt)∥2 Λi,t, (10) where xt ∈ Rn denotes the pose of the target at time t and yi,t ∈ Rm denotes robot i’s observation of the target at time t, over a duration of T + 1 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We represent the trajectory of the target with x = � x⊤ 0 , x⊤ 1 , · · · , x⊤ T �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' While the first term in the objective function corresponds to the error between the estimated state of the target at a subsequent timestep and its expected state based on a model of its dynamics, the second term corresponds to the error between the observations collected by each robot and the expected measurement computed from the estimated state of the target, where the function hi : Rn → Rm 5 denotes the measurement model of robot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Further, the information matrices Ωt ∈ Rn×n and Λi,t ∈ Rm×m for the dynamics and measurement models, respectively, weight the contribution of each term in the objective function appropriately, reflecting prior confidence in the dynamics and measurement models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The MAP optimization problem in (10) is not separable, hence, not amenable to distributed optimization, in its current form, due to coupling in the objective function arising from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Nonetheless, we can arrive at a separable optimization problem through a fairly straightforward reformulation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We can assign a local copy of x to each robot, with ˆxi denoting robot i’s local copy of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The reformulated problem becomes minimize ˆx N � i=1 T −1 � t=0 1 N ∥ˆxi,t+1 − g(ˆxi,t)∥2 Ωt + N � i=1 T −1 � t=0 ∥yi,t − hi(ˆxi,t)∥2 Λi,t subject to ˆxi = ˆxj ∀(i, j) ∈ E, (11) where ˆx = � ˆx⊤ 1 , ˆx⊤ 2 , · · · , ˆx⊤ N �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Following this reformulation, distributed optimization algorithms can be applied to compute an estimate of the trajectory of the target from (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Multi-Robot Task Assignment In the multi-robot task assignment problem, we seek an optimal assignment of N robots to M tasks such that the total cost incurred in completing the specified tasks is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, we note that many task assignment problems consist of an equal number of tasks and robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The standard task assignment problem has been studied extensively and is typically solved using the Hungarian method [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, optimization-based methods have emerged as a competitive approach due to their amenability to task assignment problems with a diverse set of additional constraints, encoding individual preferences or other relevant problem information, making them a general-purpose approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The task assignment problem can be represented as a weighted bipartite graph: a graph whose vertices can be divide into two sets where no two nodes within a given set share an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Further, each edge in the graph has an associated weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In task assignment problems, the edge weight ci,j represents the cost of assigning robot i to task j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Figure 3 depicts a task assignment problem represented by a weighted bipartite graph, with three robots and three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each robot knows its task preferences only and does not know the task preferences of other robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Equivalently, the task assignment problem can be formulated as an integer optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Many optimization-based methods solve a relaxation of the integer optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Generally, in problems with linear objective functions and affine constraints, these optimization-based methods are guaranteed to yield an optimal task assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The associated relaxed optimization Tasks Task 𝑗 Robots Robot 𝑖 𝑐!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=',# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A multi-robot task assignment problem represented as a bipartite graph, with three (Fetch) robots and three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' An edge with weight ci,j between robot i and task j signifies the cost incurred by robot i if it performs task j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In many problems, each robot’s task preferences (edge weights) is neither known by other robots nor accessible to these robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' problem takes the form minimize x N � i=1 c⊤ i xi subject to N � i=1 xi = 1M 1⊤ Mxi = 1 0 ≤ x ≤ 1, (12) where xi ∈ RM denotes the optimization variable of robot i, representing its task assignment and x = [x1, x2, · · · , xN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Although the objective function of (12) is separable, the optimization problem is not separable due to coupling of the optimization variables arising in the first constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We can obtain a separable problem, amenable to distributed optimization, by assigning a local copy of x to each robot, resulting in the problem minimize ˆx N � i=1 c⊤ i ˆxi,i subject to N � i=1 ˆxi,i = 1M 1⊤ M ˆxi,i = 1 0 ≤ ˆxi ≤ 1 ∀i ∈ V ˆxi = ˆxj ∀(i, j) ∈ E (13) where ˆxi ∈ RM×N denotes robot i’s local copy of x and ˆx = [ˆx0, ˆx1, · · · , ˆxN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Although the reformulation in (13) is simple, it does not scale efficiently with the number of robots and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A more efficient reformulation can be obtained by considering the dual formulation of the task assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For brevity, we omit a discussion of this approach in this paper and refer readers to [19], [20], [21] where this reformulation scheme is discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Collaborative Planning, Control, and Manipulation Generally, in collaborative planning problems, we seek to compute state and control input trajectories that enable a group of robots to reach a desired state configuration from a specified initial state, while minimizing a trajectory cost and freight100 : fetchrobatis6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A multi-robot manipulation problem, with three quadrotors collabora- tively manipulating a load rigidly attached to each quadrotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The dashed-line represents the reference trajectory for manipulating the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' without colliding with other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The related multi-robot control problem involves computing a sequence of control inputs that enables a group of robots to track a desired reference trajectory or achieve some specified task such as manipulating an object collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Figure 4 shows a collaborative manipulation problem where three quadrotors move an object collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The dashed-line represents the reference trajectory for manipulating the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Collaborative multi-robot planning, control, and manipulation problems have been well-studied, with a broad variety of methods devised for these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Among these methods, receding horizon or model predictive control (MPC) approaches have received notable attention due to their flexibility in encoding complex problem constraint and objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In MPC approaches, these multi-robot problems are formulated as optimization problems over a finite time duration at each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The resulting optimization problem is solved to obtain a sequence of control inputs over the specified time duration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' however, only the initial control input is applied by each robot at the current timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' At the next timestep, a new optimization problem is formulated, from which a new sequence of control inputs is computed to obtain a new control input for that timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This process is repeated until completion of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' At time t, the associated MPC optimization problem has the form minimize x,u N � i=1 fi(x, u) subject to g(x, u) = 0 h(x, u) ≤ 0 xi,0 = ¯xi ∀i ∈ V (14) where xi ∈ Rni denotes robot i’s state trajectory, ui ∈ Rmi de- notes its control input trajectory, and x = � x⊤ 1 , x⊤ 2 , · · · , x⊤ N �⊤ with u = � u⊤ 1 , u⊤ 2 , · · · , u⊤ N �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The objective function of robot i, fi : R¯n × R ¯m → R, is often quadratic, given by fi(x, u) = (xi − ˜xi)⊤Qi(xi − ˜xi) + (ui − ˜ui)⊤Ri(ui − ˜ui), (15) where ˜xi and ˜ui denote the reference state and control input trajectory, respectively, Qi ∈ Rni×ni and Ri ∈ Rmi×mi denote the associated weight matrices for the terms in the objective function, ¯n = �N i=1 ni, and ¯m = �N i=1 mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The dynamics function of the robots is encoded in g : R¯n × R ¯m → R¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Fur- ther, other equality constraints can be encoded in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Inequality constraints, such as collision-avoidance constraints and other state or control input feasibility constraints, are encoded in h : R¯n × R ¯m → Rl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, the first state variable of each agent is constrained to be equal to its initial state, denoted by ¯xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In each instance of the MPC optimization problem, the initial state ¯xi of robot i is specified as its current state at that timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Note that the MPC optimization problem in (14) is not generally separable, depending on the equality and inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, a separable form of the problem can always be obtained by introducing local copies of the optimization variables that are coupled in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The functions g and h can also encode complementarity constraints for manipulation and locomotion problems that involve making and breaking rigid body contact [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In the extreme case, where the optimization variables are coupled in the objective function and equality and inequality constraints in (14), a suitable reformulation takes the form minimize ˆx,ˆu N � i=1 fi(ˆxi, ˆui) subject to g(ˆxi, ˆui) = 0 ∀i ∈ V h(ˆxi, ˆui) ≤ 0 ∀i ∈ V φi(ˆxi) = ¯xi ∀i ∈ V ˆxi = ˆxj ∀(i, j) ∈ E, (16) where the function φi outputs the first state variable corre- sponding to robot i, given the input ˆxi, which denotes robot i’s local copy of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Similarly, ˆui denotes robot i’s local copy of u, with ˆx = � ˆx⊤ 1 , ˆx⊤ 2 , · · · , ˆx⊤ N �⊤ and ˆu = � ˆu⊤ 1 , ˆu⊤ 2 , · · · , ˆu⊤ N �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed optimization algorithms [5], [23], [24] can be employed to solve the resulting MPC optimization problem in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Multi-Robot Learning Multi-robot learning entails the application of deep learning methods to approximate functions from data to solve multi- robot tasks, such as object detection, visual place recognition, monocular depth estimation, 3D mapping, and multi-robot rein- forcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Consider a general multi-robot supervised learning problem where we aim to minimize a loss function over labeled data collected by all the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We can write this as min θ N � i=1 � (xij,yij)∈Di l(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' θ)), where l(·, ·) is the loss function, (xij, yij) is data point j collected by robot i with feature vector xij and label yij, Di is the set of data collected by robot i, θ are the neural network weights, and f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' θ) is the neural network parameterized function we desire to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' By creating local copies of the neural network weights θi and adding consensus constraints θi = θj, we can put problem in the form (5), so it is amenable to distributed optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We stress that this problem encompasses a large majority of problems in 7 𝑎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 𝑜!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Observation 𝑜" 𝑜# 𝑎# Action 𝑎" Robot 𝑖 Robot 𝑗 Robot 𝑘 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In multi-robot reinforcement learning problems, a group of robots compute a control policy from experience by making sequential decisions while interacting with their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each robot takes an action and receives an observation (and a reward), which provides information on its performance in accomplishing a specified task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' See [25] for an ADMM-based distributed optimization approach to solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Beyond supervised learning, many multi-robot learning problems are formulated within the framework of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In these problems, the robots learn a control policy by interacting with their environments by making sequential decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The underlying control policy, which drives these sequential decisions, is iteratively updated to optimize the performance of all agents on a specified objective using the information gathered by each robot during its interaction with its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Figure 5 illustrates the reinforcement learning paradigm, where a group of robots learn from experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each robot takes an action and receives an observation (and a reward), which provides information on the performance of its current control policy in achieving its specified objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Reinforcement learning approaches can be broadly cate- gorized into value-based methods and policy-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Value-based methods seek to compute an estimate of the optimal action-value function — the Q-function — which represents the expected discounted reward starting from a given state and taking a given action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' An optimal policy can be extracted from the estimated Q-function by selecting the action that maximizes the value of the Q-function at a specified state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In deep value-based methods, deep neural networks are utilized in approximating the Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In contrast, policy-based methods seek to find an optimal policy by directly searching over the space of policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In deep policy-based methods, the control policy is parameterized using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' the agents seek to maximize the expected infinite- horizon discounted cumulative reward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' which is posed as the optimization problem maximize θ Eπθ � �� t≥0 γt N � i=1 Ri(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t) | si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='0 = ¯si � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (17) where πθ denotes the control policy parameterized by θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' γ ∈ R denotes the discount factor (γ ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t denotes the state of robot i at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='t denotes its action at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ¯si denotes its initial state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Ri : Si × Ai → R denotes the reward function of robot i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' and N denotes the number of robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The optimization problem in (17) is not separable in its current form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, due to the linearity of the expectation operator, the optimization problem in (17) can be equivalently expressed as maximize ˆθ1,··· ,ˆθN N � i=1 Eπˆθi � �� t≥0 γtRi(si,t, ai,t) | si,0 = ¯si � � subject to ˆθi = ˆθj ∀(i, j) ∈ E, (18) which is separable among the N robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Hence, the resulting problem can be readily solved using distributed optimization algorithms for reinforcement learning problems, such as distributed Q-learning and distributed actor-critic methods [26], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' CLASSES OF DISTRIBUTED OPTIMIZATION ALGORITHMS In this section, we categorize distributed optimization al- gorithms into three broad classes — Distributed First-Order Methods, Distributed Sequential Convex Programming, and Alternating Direction Method of Multipliers — and provide a brief overview of each category, by considering a representative distributed algorithm within each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In the subsequent discussion, we consider the separable optimization problem in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In the tutorial spirit of this paper, we first consider problems without the local equality and inequality constraint functions, gi and hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In the second paper in this series, we include these constraint functions, as we give a survey of more sophisticated distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Before describing the specific algorithms that solve dis- tributed optimization problems, we first consider the general framework that all of these approaches share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each algorithm progresses over discrete iterations k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' until conver- gence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In general, each iteration consists of a communication step and a computation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Besides assuming that each robot has the sole capability of evaluating its local objective function fi, we also distinguish between the “internal” variables P(k) i that the robot computes at each iteration k and the “communicated” variables Q(k) i that the robot communicates to its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each algorithm also involves parameters R(k) i , which generally require coordination among all of the robots but can typically be assigned before deployment of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In distributed optimization, all the robots seek to collectively minimize the joint objective function in (5) while achieving consensus on a common set of minimizing optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this paper, we distinguish between two distinct perspectives on how consensus between the robots is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed first-order methods and distributed sequential convex programming methods enforce the consensus (equality) constraints in (5) implicitly, while the alternating direction method of multipliers enforces these constraints explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed First-Order Algorithms Gradient decent methods have been widely applied to solve broad classes of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In general, these methods only require the computation of the gradient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=', the first derivative of the objective and constraint functions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 8 hence, these methods are also referred to as first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' When applied to (5), the updates to the optimization variable take the form x(k+1) = x(k) − α(k)∇f(x(k)) (19) where α(k) denotes a diminishing step-size and ∇f(x(k)) denotes the gradient of the objective function, given by ∇f(x) = � i∈V ∇fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (20) From (20), computation of ∇f(x) requires knowledge of the objective function of all robots, which is unavailable to any individual robot, and thus requires aggregation of this information at a central node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed First-Order (DFO) algorithms provide alternative update schemes that circumvent this underlying challenge by enabling each robot to utilize only its local gradients, while communicating with its neighbors to reach consensus on a common solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We can further categorize DFO methods into two broad subclasses: Adapt-Then-Combine (ATC) methods and Combine-Then-Adapt (CTA) methods, based on the relative order of the communication and computation procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In ATC methods, each robot updates its local optimization variable using its gradient prior to combining its local variable with that of its neighbors, with the update procedure given by x(k+1) i = � j∈Ni∪{i} wij(x(k) j − α(k)y(k) j ), (21) where x(k) j ∈ Rn denotes the local variable of neighboring robot j and y(k) j denotes j’s local gradient y(k) j = ∇fj(x(k) j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Note that each robot updates its local variable x(k+1) i using gradients from its one-hop neighborhood before communicating its local variable with its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The weight wi,j must be compatible with the underlying communication network, such that wi,j = 0 if robots i and j do not share a direct communication link, and the weighting matrix W should be a stochastic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In contrast, in CTA methods, each robot combines its local variable with that of its neighbors prior to incorporating its local gradient, yielding the update procedure x(k+1) i = � j∈Ni∪{i} wijx(k) j − α(k)y(k) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (22) In the simplest case, y(k) i = ∇fi(x(k) i ), or more generally, y(k) i = ∂fi(x(k) i ) (where ∂fi denotes the subgradient of fi), yielding the canonical distributed subgradient method [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This choice of y(k) i necessitates the use of a diminishing step-size, which is often given by α(k+1) = α(0) √ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' More sophisticated gradient tracking methods, for example DIGing [30], employ an estimate of the average gradient computed through dynamic average consensus with y(k+1) i = � j∈Ni∪{i} wijy(k) j + � ∇fi(x(k+1) i ) − ∇fi(x(k) i ) � , (23) which does not require a diminishing step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' At initialization of the algorithm, all the robots select a common step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Algorithm 1: DIGing Initialization: k ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' x(0) i ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' y(0) i = ∇fi(x(0) i ) Internal variables: P(k) i = ∅ Communicated variables: Q(k) i = � x(k) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' yk i � Parameters: R(k) i = (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' wi) do in parallel ∀i ∈ V Communicate Q(k) i to all j ∈ Ni Receive Q(k) j from all j ∈ Ni x(k+1) i = � j∈Ni∪{i} wijx(k) j − αy(k) i y(k+1) i = � j∈Ni∪{i} wijy(k) j + ∇fi(x(k+1) i ) − ∇fi(x(k) i ) k ← k + 1 while stopping criterion is not satisfied Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' robot i initializes its local variables with x(0) i ∈ Rn and y(0) i = ∂fi(x(0) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Algorithm 1 summarizes the update procedures in the distributed gradient tracking method DIGing [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Many DFO methods impose additional restrictions on the weighting matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For example, DIGing requires a doubly- stochastic weighting matrix for undirected communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed Sequential Convex Programming Sequential convex programming entails solving an op- timization problem by computing a sequence of iterates, representing the solution of a series of approximations of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Newton’s method is a prime example of a sequential convex programming method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In Newton’s method, and more generally, quasi-newton methods, we take a quadratic approximation of the objective function at an operating point x(k), resulting in ˜f(x) = f(x(k)) + ∇f(x(k))⊤(x − x(k)) + 1 2(x − x(k))⊤H(x(k))(x − x(k)), (24) where H(·) denotes the Hessian of the objective function, ∇2f, or its approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Subsequently, we compute a solution to the quadratic program, given by x(k+1) = x(k) − H � x(k)�−1∇f(˜x), (25) which requires centralized evaluation of the gradient and Hessian of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Distributed Sequential Programming enable each robot to compute a local estimate of the gradient and Hessian of the objective function, and thus allows for the local execution of the update procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We consider the NEXT algorithm [31] to illustrate this class of distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We assume that each robot uses a quadratic approximation of the optimization problem as its convex surrogate model U(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In NEXT, each robot maintains an estimate of the average gradient of the objective function, as well as an estimate of the gradient of the objective function excluding its local component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=', � j̸=i fj(xi) for 9 robot i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' At a current iterate x(k) i , robot i creates a quadratic approximation of the optimization problem, given by minimize ˜xi � ∇fi(x(k) i ) + ˜π(k) i �⊤ � ˜xi − x(k) i � + 1 2 � ˜xi − x(k) i �⊤Hi � x(k) i �� ˜xi − x(k) i � , (26) where ˜π(k) i denotes robot i’s estimate of the gradient of � j̸=i fj(xi) at x(k) i , which can be solved locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each robot computes a weighted combination of its current iterate and the solution of (26), given by the procedure z(k) i = x(k) i + α(k) � ˜x(k) i − x(k) i � , (27) where α(k) ∈ (0, 1) denotes a diminishing step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Subse- quently, robot i computes its next iterate by taking a weighted combination of its local estimate z(k) i with that of its neighbors via the procedure x(k+1) i = � j∈Ni∪{i} wijz(k) j , (28) for consensus on a common solution of the original optimiza- tion problem, where the weight wi,j must be compatible with the underlying communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, robot i updates its estimates of the average gradient of the objective function, denoted by yi, using dynamic average consensus, given by y(k+1) i = � j∈Ni∪{i} wijy(k) j + � ∇fi(x(k+1) i ) − ∇fi(x(k) i ) � , (29) as well as ˜π(k) i using the procedure ˜π(k+1) i = N · y(k+1) i − ∇fi(x(k+1) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (30) Each agent initializes its local variables with x(0) i ∈ Rn, y(0) i = ∇fi(x(0) i ), and ˜π(k+1) i = Ny(0) i − ∇fi(x(0) i ), prior to executing the above update procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Algorithm 2 summa- rizes the update procedures in NEXT [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Alternating Direction Method of Multipliers The alternating direction method of multipliers (ADMM) belongs to the class of optimization algorithms referred to as the method of multipliers (or augmented Lagrangian methods), which compute a primal-dual solution pair of a given optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The method of multipliers proceeds in an alternating fashion: the primal iterates are updated as minimizers of the augmented Lagrangian, and subsequently, the dual iterates are updated via dual (gradient) ascent on the augmented Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The procedure continues iteratively until convergence or termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The augmented Lagrangian of the problem in (5) (with only the consensus constraints) is given by La(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' q) = N � i=1 fi(xi) + N � i=1 � j∈Ni � q⊤ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j(xi − xj) + ρ 2∥xi − xj∥2 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (31) Algorithm 2: NEXT Initialization: k ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' x(0) i ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' y(0) i = ∇fi(x(0) i ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ˜π(k+1) i = Ny(0) i − ∇fi(x(0) i ) Internal variables: Pi = � x(k) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ˜x(k) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ˜π(k) i � Communicated variables: Q(k) i = � z(k) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' y(k) i � Parameters: R(k) i = � α(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' U(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' K � do in parallel ∀i ∈ V ˜x(k) i = argmin x∈K U � x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' x(k) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ˜π(k) i � z(k) i = x(k) i + α(k) � ˜x(k) i − x(k) i � Communicate Q(k) i to all j ∈ Ni Receive Q(k) j from all j ∈ Ni x(k+1) i = � j∈Ni∪{i} wijz(k) j y(k+1) i = � j∈Ni∪{i} wijy(k) j + � ∇fi(x(k+1) i ) − ∇fi(x(k) i ) � ˜π(k+1) i = N · y(k+1) i − ∇fi(x(k+1) i ) k ← k + 1 while stopping criterion is not satisfied where qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j represents a dual variable for the consensus con- straints between robots i and j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' q = � q⊤ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' j) ∈ E �⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' and x = � x⊤ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' x⊤ 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' x⊤ N �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The parameter ρ > 0 represents a penalty term on the violations of the consensus constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Generally, the method of multipliers computes the minimizer of the augmented Lagrangian with respect to the joint set of op- timization variables, which hinders distributed computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In contrast, in the alternating direction method of multipliers, the minimization procedure is performed block-component-wise, enabling parallel, distributed computation of the minimization subproblem in the consensus problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, many ADMM algorithms still require some centralized computation, rendering them not fully-distributed in multi-robot mesh network sense that we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We focus here on ADMM algorithms that are distributed over robots in a mesh network, with each robot executing the same set of distributed steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We specifically consider the consensus alternating direction method of multipliers (C-ADMM) [32] as a representative algorithm within this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' C-ADMM introduces auxiliary optimization variables into the consensus constraints in (5) to enable fully-distributed update procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The primal update procedure of robot i takes the form x(k+1) i = argmin xi � fi(xi) + x⊤ i y(k) i + ρ � j∈Ni ����xi − 1 2 � x(k) i + x(k) j ����� 2 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (32) 10 Algorithm 3: C-ADMM Initialization: k ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' x(0) i ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' y(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='Internal variables: P(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='= y(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='Communicated variables: Q(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='= x(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='Parameters: R(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='= ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='do in parallel ∀i ∈ V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='x(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='= argmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='xi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='fi(xi) + x⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i y(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='+ ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j∈Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='����xi − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='x(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='+ x(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='Communicate Q(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='to all j ∈ Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='Receive Q(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='from all j ∈ Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='y(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='= y(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='+ ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j∈Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='x(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='− x(k+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='k ← k + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='while stopping criterion is not satisfied ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='which only requires information locally available to robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' including information received from its neighbors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=', xk j , ∀j ∈ Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' As a result, this procedure can be executed locally by each agent, in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' After communicating with its neighbors, each robot updates its local dual variable using the procedure y(k+1) i = y(k) i + ρ � j∈Ni � x(k+1) i − x(k+1) j � , (33) where yi denotes the composite dual variable of robot i, corresponding to the consensus constraints between robot i and its neighbors, which is initialized to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Algorithm 3 summarizes the update procedures in C-ADMM [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' DISTRIBUTED MULTI-DRONE VEHICLE TRACKING: A CASE STUDY Many robotics problems have a distributed structure, al- though this structure might not be immediately apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In many cases, applying distributed optimization methods requires reformulating the original problem into a separable form that allows for distributed computation of the problem variables locally by each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This reformulation often involves the introduction of additional problem variables local to each robot with an associated set of constraints relating the local variables between the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We illustrate this procedure using multi- drone vehicle target tracking as a case study in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We note that the same principles apply to a broad class of robotics problems as we have outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, we implement the distributed optimization algorithm C-ADMM on hardware, to demonstrate the deployment of distributed optimization algorithms on hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Simulation Study In this simulation, we consider a distributed multi-drone vehicle target tracking problem in which robots connected by a communication graph, G = (V, E), each record range- limited linear measurements of a moving target, and seek to collectively estimate the target’s entire trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We assume that each drone can communicate locally with nearby drones over the undirected communication graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The drones all share a linear model of the target’s dynamics as xt+1 = Atxt + wt, (34) where xt ∈ R4 represents the position and velocity of the target in some global frame at time t, At is the dynamics matrix associated with a linear model of the target’s dynamics, and wt ∼ N(0, Qt) represents process noise (including the unknown control inputs to the target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Restricting our case study to a linear target model in this tutorial ensures that the underlying optimization problem is convex, leading to strong convergence guarantees and robust numerical properties for our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A more expressive nonlinear model can also be used, but this requires a more sophisticated distributed optimization algorithm with more challenging numerical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' At every time-step when the target is sufficiently close to a drone i (which we denote by t ∈ Ti), that robot collects an observation according to the linear measurement model yi,t = Ci,txt + vi,t , (35) where yi,t ∈ R2 is a positional measurement, Ci,t is the measurement matrix of drone i, and vi,t ∼ N(0, Ri,t) is measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We again assume a linear measurement model to keep this case study as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A nonlinear model can also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' All of the drones have the same model for the prior distribu- tion of the initial state of the target N(¯x0, ¯P0), where ¯x0 ∈ R4 denotes the mean and ¯P0 ∈ R4×4 denotes the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The global cost function is of the form f(x) =∥x0 − ¯x0∥2 ¯ P −1 0 + T −1 � t=1 ∥xt+1 − Atxt∥2 Q−1 t + � i∈V � t∈Ti ∥yi,t − Ci,txt∥2 R−1, (36) while the local cost function for drone i is fi(x) = 1 N ∥x0 − ¯x0∥2 ¯ P −1 0 + T −1 � t=1 1 N ∥xt+1 − Atxt∥2 Q−1 t + � t∈Ti ∥yi,t − Ci,txt∥2 R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' (37) In our results, we consider only a batch solution to the problem (finding the full trajectory of the target given each robot’s full set of measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Methods for performing the estimate in real-time through filtering and smoothing steps have been well studied, both in the centralized and distributed case [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' An extended version of this multi-robot tracking problem is solved with distributed optimization in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' A rendering of a representative instance of this multi-robot tracking problem is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In Figures 6 and 7, several distributed optimization algo- rithms are compared on an instance of the distributed multi- drone vehicle tracking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For this problem instance, 10 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Mean Square Error (MSE) per iteration on a distributed multi-drone vehicle target tracking problem with N = 10 and n = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' simulated drones seek to estimate the target’s trajectory over 16 time steps resulting in a decision variable dimension of n = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We compare four distributed optimization methods which we consider to be representative of the taxonomic classes outlined in the sections above: C-ADMM [32], EXTRA [34], DIGing [30], and NEXT-Q [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Figure 6 shows that C-ADMM and EXTRA have similar fast convergence rates per iteration while DIGing and NEXT-Q are 4 and 15 times slower respectively to converge below an MSE of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The step-size hyperparameters for each method are computed by Golden Section Search (GSS) (for NEXT-Q, which uses a two parameter decreasing step-size, we fix one according to the values recommended in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We note that tuning is essential for achieving robust and efficient convergence with most distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Figure 7 shows the sensitivity of these methods to variation in step-size, and highlights that three of the methods (all except C-ADMM) become divergent for certain subsets of the tested hyperparameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' While C-ADMM seems to be the most effective algorithm in this problem instance, we note that other algorithms may have properties that are advantageous in other instances of this problem or other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' For example, C-ADMM is known to require tight synchronization among the robots (or the computing nodes, more generally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' If a robot misses a message or if robot clocks are not precisely synchronized, C-ADMM can perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' First order methods, such as DIGing, tend to be more robust to asynchronicity, for example, and modifying these algorithms totolerate real-world challenges such as asynchronicity remains an area of active research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We discuss such issues in the second paper in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Hardware Implementation In this section, we discuss our implementation of the C-ADMM algorithm on hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Each robot is equipped with local computational resources and communication hardware Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Step-size hyperparameter sensitivity sweep on a distributed multi-drone vehicle target tracking problem for N = 10 and n = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The dashed lines are thresholds for divergence (top) and convergence (bottom) in terms of MSE after 104 decision variable updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' necessary for peer-to-peer communication with other neighbor- ing robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In the following discussion, we provide details of the hardware platform, the underlying communication network between robots, and the optimization problem considered in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We consider the linear least-squares optimization problem min p N � i=1 (Gip − zi)⊤Mi(Gip − zi), (38) with the optimization variable p ∈ R32, Gi ∈ Rmi×32, Mi ∈ Rmi×mi, zi ∈ Rmi, and N = 3 robots, where mi de- pends on the number of measurements available to robot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In this experiment, we have m1 = 3268, m2 = 5422, and m3 = 3528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We implement C-ADMM to solve the problem, with a state size consisting of 32 floating-point variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The core communication infrastructure that we use are Digi XBee DigiMesh 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content='4 radio frequency mesh networking modules which allow for peer-to-peer communication between robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Local computation for each robot is performed using Raspberry Pi 4B single board computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The lower level mesh network is managed by the DigiMesh software, and we interact with it through XBee Python Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We utilize the neighbor discovery Application Programming Interface (API) provided by Digi International to enable each robot to identify other neighboring robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' This approach resulted in a fully-connected communication network, con- sidering the XBee radios have an indoor range of up to 90m and an outdoor range of up to 1500m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The XBee modules used in our experiments have a maximum payload size of 92 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, the local variable of each robot in our experiment consists of 32 floating-point variables, which exceeds the maximum payload size that can be transmitted by the XBee radios at each broadcast round, presenting a communication challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' To overcome this challenge, we break up the local variables into a series of packets of size 12 0 50 100 150 200 250 Number of Iterations 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 ‖x − x⋆‖2 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Convergence of the iterates computed by each robot using C-ADMM, implemented on hardware, on the optimization problem with three robots in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The convergence errors of all the robots overlap in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' 92 bytes and perform multiple broadcast rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We note, however, that other approaches could be employed to overcome the limited communication bandwidth of the XBee radios, including utilizing quantization-based distributed optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We set the penalty parameter in C-ADMM to a value of 5 and do not perform a comprehensive search for the penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In our experiments, we noticed that this value of the penalty parameter provided suitable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Noting that C-ADMM requires synchronous updates, we ensure that the local clocks of all robots remain synchronized in our experiments using a barrier strategy, which prevents each robot from advancing to the next iteration of C-ADMM until all other robots have completed the current iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In Figure 8, we show the convergence error between the iterates of each robot and the global solution, which is obtained by aggregating the local data of all robots and then computing the solution centrally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' The convergence errors of all the robots iterates overlap in the figure, with the error decreasing below 10−5 within 250 iterations, showing convergence of the local iterates of each robot to the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' CONCLUSION In this tutorial, we have demonstrated that a number of canonical problems in multi-robot systems can be formulated and solved through the framework of distributed optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' We have identified three broad classes of distributed optimiza- tion algorithms: distributed first-order methods, distributed sequential convex programming methods, and the alternat- ing direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Further, we have described the optimization techniques employed by the algorithms within each category, providing a representative algorithm for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' In addition, we have demonstrated the application of distributed optimization in simulation, on a distributed multi-drone vehicle tracking problems, and on hardware, showing the effectiveness of distributed optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' However, important challenges remain in develop- ing distributed algorithms for constrained, non-convex robotics problems, and algorithms tailored to the limited computation and communication resources of robot platforms, which we discuss in greater detail in the second paper in this series [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank Siddharth Tanwar for his help in performing the hardware experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFIT4oBgHgl3EQfnSu4/content/2301.11313v1.pdf'} +page_content=' Rockafellar, “Monotone operators and the proximal point 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sha256:0be2372b3c9f35a6d5895de88b5def167af8535dcf0cd197e93c4cf7f59e59b1 +size 6160429 diff --git a/h9E4T4oBgHgl3EQfrw0X/content/tmp_files/2301.05210v1.pdf.txt b/h9E4T4oBgHgl3EQfrw0X/content/tmp_files/2301.05210v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dd5621636bd7f1f780b6d9ba36450ca24a2243d --- /dev/null +++ b/h9E4T4oBgHgl3EQfrw0X/content/tmp_files/2301.05210v1.pdf.txt @@ -0,0 +1,2210 @@ +MNRAS 000, 1–14 (2022) +Preprint 13 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The satellite population around luminous red galaxies in the 25 𝑑𝑒𝑔2 DESI +Legacy Imaging Surveys Early Data Release +Melinda Townsend,1★ and Gregory Rudnick1† +1University of Kansas, Department of Physics and Astronomy, 1082 Malott,1251 Wescoe Hall Dr., Lawrence, KS 66045 +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Luminous Red Galaxies, or LRGs, are representative of the most massive galaxies and were originally selected in the Sloan +Digital Sky Survey as good tracers of large scale structure. They are dominated by by uniformly old stellar populations, have +low star formation rates, early type morphologies, and little cold gas. Despite having old stellar populations and little in situ star +formation, studies have shown that they have grown their stellar mass since z=1, implying that they grow predominantly via the +accretion of satellites. Tests of this picture have been limited because of the lack of deep imaging data sets that both covers a large +enough area of the sky to contain substantial numbers of LRGs and that also is deep enough to detect faint satellites. We use the +25 deg2 Early Data Release (EDR) of the DESI Legacy Imaging Surveys to characterize the satellite galaxy population of LRGs +out to z=0.65. The DESI Legacy Imaging Surveys are comprised of 𝑔𝑟𝑧 imaging to 2-2.5 mag deeper than SDSS and with better +image quality. We use a new statistical background technique to identify excess populations of putative satellite galaxies around +1823 LRGs at 0.2 < 𝑧 < 0.65. In three redshift and luminosity bins we measure the numbers of satellite galaxies and their +𝑟 − 𝑧 color distribution down to rest-frame 𝑔-band luminosity limits at least 3.6 times fainter than 𝐿∗. In addition, we develop a +forward modeling technique and apply it to constrain the mean number of satellites in each of our redshift and luminosity bins. +Finally, we use these estimates to determine the amount of stellar mass growth in LRGs down to the local Universe. +Key words: galaxies: evolution – galaxies: elliptical and lenticular, cD – Galaxy: general +1 +Introduction +The method by which massive galaxies accumulate their mass is an +open question in galaxy evolution. Early-type galaxies (ETGs) have +low specific star formation rates, old stellar populations, and very +little cold gas (e.g. Moustakas et al. 2013). A homogeneous pop- +ulation of early-type galaxies are Luminous Red Galaxies (LRGs). +Originally selected as good tracers of large scale structure (Eisenstein +et al. 2001), subsequent investigations have shown them to be among +the most massive galaxies in the universe, dominated by uniformly +old stellar populations (Tojeiro & Percival 2011). Despite this, the +stellar mass of LRGs has grown by 50% since z=0.9, implying that +they grow mostly through the accretion of passive satellites (Cool +et al. 2008). Multiple studies have also shown that the spectra of +LRGs indicate that those galaxies have undergone passive fading +(e.g. (Cool et al. 2008; Banerji et al. 2010)). However, some stud- +ies find that only bright LRGs evolve passively while fainter LRGs +have more extended star formation histories (SFG; Tojeiro & Percival +2011). +It is now well accepted that massive galaxies grow hierarchically +through the successive effect of many mergers coupled with in situ +growth by star formation (e.g. Frenk & White 1991). There is over- +whelming observational support for this overall picture, with some of +★ E-mail: mtownsend@ku.edu +† E-mail: grudnick@ku.edu +the most direct evidence coming from the observations that massive +galaxies at 𝑧 > 0.8 have significantly smaller sizes and than similarly +massive galaxies in the local universe (Daddi et al. 2005; Trujillo +et al. 2006; van der Wel et al. 2008; Buitrago et al. 2008; van der +Wel et al. 2008; van Dokkum et al. 2008; Hopkins et al. 2009; van +Dokkum et al. 2010; Newman et al. 2012; van der Wel et al. 2014). +For example, Trujillo et al. (2006) found that massive galaxies at +1.4 < 𝑧 < 1.7 were at least a factor of four times smaller in the rest- +frame 𝑉-band than local counterparts of the same stellar mass, and +that the the stellar density of these objects are at least 60 times larger +than present-day massive ellipticals, indicating that they must have +grown predominantly through dry mergers and Daddi et al. (2005) +hypothesized that the size growth came from the accretion of satellite +galaxies. +Likewise, Man et al. (2016) found that, to explain the ob- +served number density evolution of massive galaxies, minor mergers +are a necessary component to bring compact quiescent ellipticals +into agreement with the stellar mass-size relation. Accretion-based +growth of passive galaxies leads to "inside-out" growth, where mas- +sive galaxies grow via a series of minor mergers and a build-up of +extended stellar halos (van Dokkum et al. 2010). Using luminosity +functions of LRGs and their satellites, Tal et al. (2012) determined +that the mass ratio of LRGs to satellites is 4:1, making mass growth +through major mergers unlikley. This is consistent with the findings +in Bezanson et al. (2009), which find that central stellar densities of +high redshift ellipticals are only a factor of a few higher when com- +© 2022 The Authors +arXiv:2301.05210v1 [astro-ph.GA] 12 Jan 2023 + +2 +M. E. Townsend et al. +pared to local ellipticals even though the total stellar density of high +redshift ellipticals are orders of magnitude higher than their local +counterparts, indicating that mass is being deposited in the outskirts +of the galaxy through minor mergers. van de Sande et al. (2013) +found that for fixed dynamical mass of massive quiescent galaxies +from z ∼ 2, the mass density within one effective radius decreases +by a factor of 20 while within a fixed physical radius of 1 kpc the +mass density decreases by a factor of ∼ 2, which is consistent with +inside-out growth through minor mergers. +These observational results are supported by simulations of galaxy +growth. Naab et al. (2010), using a high-resolution hydrodynamical +cosmological simulation, shows that the accretion of weakly bound +material - or minor mergers - can cause the radius of a massive +spheroidal galaxy to increase as the square of the mass, whereas major +mergers would cause the radius to increase at a linear rate. This result +echos the observations and implies a cause; namely, that early-type +galaxies are more compact at earlier times and grow less compact at +low redshift because of the accretion of low-mass satellites. +The rest-frame 𝐵-band luminsosity function of red galaxies also +points to mergers as a mechanism to build up mass in massive sys- +tems. Bell et al. (2004) presented the rest-frame colors and lumi- +nosities of ∼ 25,000 galaxies in 0.2 < z < 1.1 from the 0.78 deg2 +Classifying Objects by Medium-Band Observations in 17 Filters +(COMBO-17) survey. They found that the 𝐵-band luminosity den- +sity does not evolve significantly in this redshift range, which, when +coupled with the passive fading of the galaxy stellar populations, +implies that there has been a build-up of stellar mass in the non-star +forming population by a factor of 2 since z ∼ 1. Faber et al. (2007) +came to a similar result in their study that compares the luminosity +functions from Willmer et al. (2006) of red and blue galaxies out +to z ∼ 1 of the DEEP2 and COMBO-17 surveys. They also found a +virtually constant 𝐵-band luminosity density for red galaxies since z +∼ 1, while the luminosity density of blue galaxies falls by ∼0.6 dex. +They argue that dry mergers are involved in building up present day +massive ellipticals. Similarly, Brown et al. (2007) used the luminos- +ity density and number density of galaxies to find that the amount of +stellar mass contained in 𝐿∗ red galaxies has doubled since z = 1, +but that the stellar mass in 4𝐿∗ red galaxies evolved more slowly. +In contrast, De Propris et al. (2010) derived an upper limit for a +dry merger rate in their measurement of the fraction of LRGs at 0.45 +< z < 0.65 in dynamically close pairs taken from the 2dF-SDSS +LRG and QSO (2SLAQ) redshift survey. They find that minor dry +mergers (a luminosity ratio 1:4 or higher) are unimportant to the +mass build-up of the red sequence at z < 0.7. This holds at higher +redshift, as well; mergers are found to not be the dominant channel +for stellar mass build-up in early-type galaxies out to z < 0.8 (Cimatti +et al. 2006) and for quiescent galaxies out to z < 1 (Moustakas et al. +2013). Scarlata et al. (2007) determined that dry mergers are likely +not a significant contributor to the build-up of massive ETGs, and +that the most massive of these galaxies were already assembled 8 Gyr +ago (although fainter ETGs keep assembling mass from z = 0.7 to +the present (Tojeiro & Percival 2011). Banerji et al. (2010) finds that +the stellar mass function for LRGs with 𝑀★ > 3 × 1011M⊙ shows +little evolution between 0.4 < z < 0.8, suggesting that most massive +systems were in place by z = 0.8 (see also Huertas-Company et al. +2016). +There are theoretical reasons to believe that understanding +accretion-based growth of massive ellipticals are important to un- +derstanding how galaxies evolve. De Lucia & Blaizot (2007) found +that of z = 0 brightest cluster galaxies (BCGs) in the Millennium +Simulation, only 10 percent were formed before z ∼ 1, and half were +assembled after z ∼ 0.5. They also find that 50 percent of the stars +found in BCGs were already formed by z ∼ 5. Old stellar populations +coupled with the late assembly times of these galaxies indicate that +the BCGs in their sample gained most of their mass through accre- +tion of satellite galaxies. Using mock catalogs constructed from the +halo occupation distribution framework and comparing to data from +the Boötes field for galaxies between 0.5 < z < 0.9, White et al. +(2007) found that their models overpredict the number of satellites +that populate massive halos and interpret this as evidence that mas- +sive satellite galaxies are merging or otherwise being disrupted by the +central galaxy in that redshift range. Building off this study, Brown +et al. (2008) measures the halo occupation distribution for red galax- +ies in the Boötes field and found that, while most massive galaxy +stellar mass growth occurs prior to z = 1, massive galaxy growth +continues and that a typical central galaxy grows by 30 percent from +z < 1. +Although there is not uniform agreement, there are many indi- +cations in both observations and simulations that merger events are +responsible for the mass build up of massive ellipticals, but how this +growth depends on redshift and primary galaxy mass is inconclu- +sive. In addition, it is unclear what role is played by faint satellites. +Most previous studies only probe satellites that are relatively mas- +sive. De Propris et al. (2010) and Bell et al. (2006) focused on the +mass build up of early-type galaxies through major mergers. Bundy +et al. (2009) probed satellites out to z ∼ 1.4 to log(M∗/M⊙) = 9.8, +but for a combined field of less than a degree. +It has been difficult to study faint satellites because of the dearth +of data that can identify those faint satellites while at the same time +encompassing a large sample of giant ellipticals. This situation is +changing with the arrival of the DESI Legacy Imaging Surveys +(Legacy Survey) (Dey et al. 2018). While large surveys have rev- +olutionized the field of galaxy evolution, there has not until recently +been data sets that both cover a large enough area of the sky to con- +tain a substantial number of luminous ellipticals and deep enough to +detect faint satellites out to intermediate redshifts. Surveys like SDSS +are too shallow to see very faint objects around luminous galaxies, +and surveys like the NOAO Deep Wide-Field Survey only cover a +small fraction of the sky. In this paper, we combine the deep photom- +etry of the Legacy Survey and the spectroscopic data from SDSS to +characterize the satellite population around SDSS-identified LRGs +2 to 2.5 magnitudes deeper than SDSS. By combining these two +surveys, we solve the problem of shallow survey depth and limited +sky coverage. We use the two surveys in the following way: We use +SDSS spectroscopy to select LRGs and the Legacy Survey imaging +to detect faint candidate satellite galaxies. We use statistical back- +ground techniques to isolate likely satellites and study the abundance +and properties of those satellites. +This paper is structured as follows: Sec. 2 describes the data and +sample selection, Sec. 3 describes our analysis method, we present +our results in Sec. 4, and discuss these results in Sec. 5. In Sec. 6 we +present our summary and conclusions. We assume H0 = 69.6, Ω𝑚 = +0.286, and ΩΛ = 0.714 Bennett et al. (2014)). +2 +Data Description +In this study we use SDSS-identified Luminous Red Galaxies using +spectroscopic redshifts of SDSS and photometry from the DESI +Legacy Imaging Surveys. +2.1 +SDSS Luminous Red Galaxies +The sample of LRGs were identified in the Baryon Oscillation Spec- +troscopic Survey (BOSS), part of the SDSS-III project. The BOSS +LRGs are divided into a low redshift sample (LOWZ) and a constant- +mass sample (CMASS) (Reid et al. 2016). Both the LOWZ and +MNRAS 000, 1–14 (2022) + +LRG satellites +3 +240 +241 +242 +243 +244 +245 +246 +247 +RA (deg) +6 +7 +8 +9 +10 +11 +12 +Dec (deg) +0 +50 +100 +150 +200 +250 +counts +Figure 1. Density maps of the EDR after we eliminate Legacy Survey sources +with observed 𝑧-band magnitude ≥ 22.75. The density variations reflect true +variations in the large scale structure of galaxies. +CMASS samples were selected based on a set of color-magnitude +cuts. For the LOWZ sample, these cuts are designed to only select the +brightest and reddest galaxies at low redshift (z < 0.4) and are similar +to the SDSS- I/II Cut-I Luminous Red Galaxies. The LOWZ sample +is also approximately volume-limited over 0.2 < z < 0.4 and has a +constant space density of ∼3 × 10−4 h3Mpc−3. The CMASS sample +includes galaxies between 0.4 < z < 0.7 with an approximately con- +stant stellar mass limit over the redshift range. The CMASS sample is +also selected by a color cut, similar to SDSS-I/II Cut-II and 2SLAQ +LRGs. However, the cuts are bluer and more faint to increase the +number density of targets in the CMASS redshift range. Higher red- +shift galaxies in CMASS are isolated using (𝑔 − 𝑟) and (𝑟 −𝑖) colors. +Together, LOWZ and CMASS make a spectroscopic sample that is +80 percent complete at log10(M/M⊙) ≥ 11.6 at z < 0.61. We choose +to study LRGs out to 0.65. There are 151 LRGs between 0.61 < z < +0.65 and may bias our sample in favor of older LRGs. However, we +ran our analysis on samples with the maximum redshift of 0.65 and +0.61 and found no difference in our results. For more details on the +selection criteria for LOWZ and CMASS (see Reid et al. (2016)). +Spectroscopic redshifts for this study are from SDSS DR14. +We identify 1,823 LRGs in the EDR in the redshift range 0.2 < z +< 0.65. We tested to see if LRG-LRG pairs skew our analysis. We +explicitly removed all LRGs that had an LRG within the approximate +virial radius in redshift slices of 0.003. Of the 1,823 LRGs in this +sample, 92 were removed because they were part of an LRG-LRG +pair, leaving us with a sample of 1,731 LRGs. The original 1,823 +is our “pair” sample, because it includes LRG-LRG pairs, and the +1,731 LRGs is our “no pair” sample. We calculated the distribution of +satellite galaxies for both the “pair” and “no pair” sample and found +that they were consistent with each other to within one standard +deviation. We conclude, then, that including LRG-LRG pairs does +not alter our results. +2.2 +Legacy Survey Description +The DESI Legacy Imaging Surveys‡ are a trio of surveys that im- +age ≈ 14,000 deg2 of the extragalactic sky visible from the northern +hemisphere in three optical bands - 𝑔, 𝑟, and 𝑧 - with a combined sur- +vey footprint that is split into two contiguous regions by the galactic +plane (Dey et al. 2018). Those surveys are the Dark Energy Cam- +era Legacy Survey, the Beijing-Arizona Sky Survey, and the Mayall +𝑧-band Legacy Survey. The data used in this paper comes from the +Dark Energy Camera Legacy Survey. +The Dark Energy Camera Legacy Survey (DECaLS) utilizes the +Blanco 4-m telescope at Cerro Tololo Inter-American Observatory +using the Dark Energy Camera and covers 9,000 deg2 in both the +Northern and Southern Galactic Cap. The survey obtains optical +imaging data in the 𝑔-, 𝑟-, and 𝑧-bands, overlaps existing spec- +troscopy from the Sloan Digital Sky Survey (SDSS), and reaches +5𝜎 point source depths of 24.0, 23.4, and 22.5 in the 𝑔-, 𝑟-, and +𝑧-band, respectively, more than two magnitudes deeper than SDSS. +Source detection is performed using The Tractor§, a forward mod- +eling algorithm in which each source is modeled at the pixel level +through simultaneous fits to a set of individual images. Each source +is modeled using a set of parametric light profiles: a delta function, +deVaucouleurs r−1/4 law, exponential disk, or exponential disk plus +deVaucouleurs. See Dey et al. (2018) for a more detailed explanation. +For this study we use the photometry from the 8th data release¶. +For this study we use data from the area covered by the Legacy +Survey Early Data Release (EDR), which covers between 240 and +245 degrees in right ascension and between 6.5 and 11.5 degrees in +declination. Fig. 1 illustrates the z-band source density in the field +of study. We want to establish a uniform detection limit across the +survey area to make sure we are equally sensitive to satellites for +all LRGs. To do this, we use the 5𝜎 galaxy detection limit in the +z-band to find the 90 percent brightest galaxy depth, and we limit +our sample to galaxies brighter than this limit. We find this limit +by binning sources into equal-size pixels using HEALPix∥ (Gorski +et al. 2005; Zonca et al. 2019) and finding the median depth in each +pixel. HEALPix discretizes the surface of a sphere into equal-area, +non-overlapping tiles called pixels and astrophysical analysis can be +done on a per-pixel basis. Then we determine the 10th percentile and +use that as our magnitude limit. We find this limit to be a 𝑧-band +magnitude 22.75 mag. We also confirm that all sources detected in +the 𝑧-band are also detected in the 𝑔- and 𝑟-band. The results of this +𝑧-band magnitude cut are shown in Fig. 1. The density map for our +magnitude complete sample demonstrates the presence of true large +scale structure variations in our field. As we will describe in Sec. 3.1, +these variations motivate our use of a local background subtraction +technique. +2.3 +Rest Frame Magnitudes and Masses for LRGs +To further characterize the SDSS LRGs, we compute both the rest +frame 𝑟-band magnitudes and the stellar masses. We compute the +rest-frame magnitudes for BOSS LRGs using the photometric red- +shift code EAZY (Brammer et al. 2008), fixing each LRG at its +spectroscopic redshift and using as inputs the 𝑔𝑟𝑧 photometry from +the Legacy Surveys, as well as WISE bands 𝑊1, 𝑊2, 𝑊3, and 𝑊4. +LRG stellar masses are computed using iSEDfit from Moustakas +et al. (2013) assuming a Kroupa IMF (Kroupa 2001). iSEDfit uses +‡ https://www.legacysurvey.org/ +§ http://thetractor.org/ +¶ https://www.legacysurvey.org/dr8/ +∥ http://healpix.sourceforge.net +MNRAS 000, 1–14 (2022) + +4 +M. E. Townsend et al. +0.2 +0.3 +0.4 +0.5 +0.6 +redshift +10.0 +10.5 +11.0 +11.5 +12.0 +log10(M*) +0.2 +0.3 +0.4 +0.5 +0.6 +redshift +23 +22 +21 +20 +19 +Mr +LOWZ +CMASS +Figure 2. The plot shows the log10(M★) and absolute 𝑟-band magnitude of LRGs in our sample v. LRG redshift in the left- and right-hand panels, respectively. +LOWZ LRGs are indicated by indigo squares and CMASS LRGs are indicated by green circles. Vertical lines show the bounds of our redshift bins. +Baysian inference to extract physical properties from a galaxy’s ob- +served broadband spectral energy distribution. +Fig. 2 shows the stellar mass and r-band absolute magnitude as +a function of redshift for SDSS LRGs. The stellar mass (left plot) +remains constant with redshift, although there is some scatter into +lower stellar mass. Absolute 𝑟-band magnitude also shows scatter at +low redshift toward fainter magnitudes. We have 516 LRGs from the +LOWZ sample and 1,307 from CMASS, for a total sample of 1,823 +LRGs. Of these, 3.7 percent have stellar masses below 1011 with 19 +appearing in the LOWZ sample and 49 appearing in the CMASS +sample, and 3.3 percent have M𝑟 > -20.5 with 19 appearing in the +LOWZ sample and 41 appearing in the CMASS sample. Visual in- +spection of these outliers show that these galaxies are predominantly +red elliptical galaxies. +2.4 +Luminosity Completeness +The redshift range of our LRG sample is 0.2 < z < 0.65. We wish +to be equally complete to satellites above a given luminosity limit +over a range in redshift. We decide to split our sample into three +luminosity-complete subsamples, each spanning a different range in +redshift. We determine the corresponding luminosity limit for each +subsample using the UltraVISTA catalog from Muzzin et al. (2013), +which is an ultra-deep 𝐾𝑠-selected catalog in the COSMOS field. +The catalog covers 1.62 deg2 and has photometry from 30 bands, to +much deeper limits than the Legacy Surveys. We use this catalog to +determine the 90 percent completeness limit in the rest-frame 𝑔-band +luminosity for three redshift bins: 0.2 < z < 0.35, 0.35 < z < 0.5, +and 0.5 < z < 0.65. As we wish to determine our luminosity in the +DECam 𝑔-band filter, we use EAZY to synthesize 𝑔-band rest-frame +luminosities using the DECam filter curve (Brammer et al. 2008). +We note that we do not use the 𝑟-band because our satellites are too +faint to be detected in WISE 𝑊1 and 𝑊2 and therefore the 𝑟-band +photometry would be unconstrained. LRGs are bright enough to be +detected in WISE, so we use the 𝑟-band luminosity for them. +Using only the UltraVISTA galaxies within our redshift range, +we measure the luminosity down to which we recover 90 percent +of the sources brighter than that limit at the high end of each red- +shift bin, subject to our observed 𝑧-band magnitude limit of ≤ 22.75. +We go through this process for each redshift bin and for all Ul- +traVISTA sources, red UltraVISTA sources, and blue UltraVISTA +sources (splitting the red and blue populations at (𝑈 − 𝑉) = 1.4). +We chose the most conservative luminosity where 90 percent of the +galaxies were recovered for each redshift bin. +Fig. 3 shows the log10(𝐿𝑔/𝐿 ⊙) of the 𝑧-band magnitude ≤ 22.75 +UltraVISTA galaxies out to z = 1, with the solid vertical lines delini- +ating the boundaries of our redshift bins. The three horizontal lines +indicate the luminosity limit for each redshift bin and the colorbar +shows the observed 𝑧-band magnitude of the UltraVISTA sources. +We have three luminosity complete samples for three redshift bins: +For 0.2 < z < 0.35 the luminosity limit is log10(𝐿𝑔/𝐿 ⊙) ≥ 9.27, for +0.2 < z < 0.5 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.58, and for +0.2 < z < 0.65 the luminosity limit is log10(𝐿𝑔/𝐿 ⊙) ≥ 9.85. Notice +that the depth of the Legacy Surveys make it possible to detect faint +galaxies all the way down to log10(𝐿𝑔/𝐿 ⊙) = 9.85 even in the upper +redshift bin. In Rudnick et al. (2009), they construct luminosity func- +tions for red sequence galaxies from 0.4 < z < 0.8. At 𝑧 ∼ 0.6, they +found that M★𝑔 ∼ −21 which corresponds to log10(𝐿𝑔/𝐿 ⊙) = 10.43, +or 3.6 times higher than our limit here in our highest redshift bin. +This means at our high redshift limit we are 3.6 times lower than L*. +3 +Determination of the number of satellite galaxies +The Legacy Survey 𝑔𝑟𝑧 photometry is not sufficient to achieve precise +photometric redshfits for our candidate satellite galaxies and obtain- +ing complete spectroscopy to these depths and over even just the EDR +would be impractical. Therefore we quantify the satellite population +around LRGs by implementing a statistical background subtraction +method. We describe this method in the following subsections. +3.1 +Statistical Background Subtraction +The foundation of our statistical background subtraction method con- +sists of three parts: counting the number of near neighbors within +a defined search radius, determining how many background sources +to expect, and subtracting the second number from the first for each +LRG. +We set our search radius to correspond to the expected virial radius +(𝑅200) for our LRGs. To estimate this radius we used the stellar mass +to halo mass relation and estimated the virial radius from the halo +mass. The 75th and 25th percentile of LRGs masses are separated +MNRAS 000, 1–14 (2022) + +LRG satellites +5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +redshift +6 +7 +8 +9 +10 +11 +log10(Lg/L +) +log10(Lg/L +) = 9.27 +log10(Lg/L +) = 9.58 +log10(Lg/L +) = 9.85 +17 +18 +19 +20 +21 +22 +observed zmag +Figure 3. The log10(𝐿𝑔/𝐿⊙) of the UltraVISTA galaxies at z < 1 and with an +observed 𝑧-band magnitude brighter than 22.75. The solid lines denote our +redshift bins. Each horizontal line marks the 90 percent luminosity complete- +ness for different redshift ranges. The colorbar indicates the observed z-band +magnitude of the sources. This defines three luminosity complete subsam- +ples: for 0.2 < z < 0.35 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.27, for 0.2 +< z < 0.5 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.58, and for 0.2 < z < +0.65 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.85. +by 0.2 dex and the median mass does not change with redshift, so +we assume that LRGs have a fixed stellar mass of log10(M/M⊙) = +11.3. We use equations 2 and 11-14 in Moster et al. (2013), which +are reproduced below, where we have slightly changed the original +notation for clarity: +𝑀★ +𝑀ℎ += 2𝑁 +�� 𝑀ℎ +𝑀1 +𝛽� ++ +� 𝑀ℎ +𝑀1 +𝛾��−1 +(1) +log 𝑀1(𝑧) = 𝑀10 + 𝑀11 +𝑧 +𝑧 + 1 +(2) +𝑁(𝑧) = 𝑁10 + 𝑁11 +𝑧 +𝑧 + 1 +(3) +𝛽(𝑧) = 𝛽10 + 𝛽11 +𝑧 +𝑧 + 1 +(4) +𝛾(𝑧) = 𝛾10 + 𝛾11 +𝑧 +𝑧 + 1 +(5) +where 𝑀★ and 𝑀ℎ are the stellar and halo mass respectively, 𝑀1 is +the characteristic mass, 𝑁 is the normalization, and 𝛽 and 𝛾 are the +power law slopes. We use best fit values for 𝑀10, 𝑀11, 𝑁10, 𝑁11, +𝛽10, 𝛽11, 𝛾10, and 𝛾11 from Table 1 in Moster et al. (2013). Using +this relation we find log(𝑀ℎ/𝑀⊙) = 13.93, 13.99, and 14.04 in our +three redshift bins. We therefore adopt log(𝑀ℎ/𝑀⊙) ≈ 14.0 for all +LRGs. Using this, we estimate 𝑅200 using the critical density (𝜌𝑐) at +each epoch and the following equation: +𝑅200 = +� 4𝜋(200𝜌𝑐) +3𝑀ℎ +�1/3 +. +(6) +The corresponding values of R200 for our sample is are 0.57, 0.59, +and 0.64 Mpc in our three redshift bins. We therefore assume that the +virial radii for our LRGs can be approximated by 𝑅200 ≈ 0.6 Mpc +and we count satellites within this radius. +To count the number of near neighbors N𝑛𝑛 around each LRG, +we used the k-d tree algorithm from scikit-learn (Pedregosa et al. +2011) to find sources projected within a 600 kpc radius, converted to +a unique angular distance for each LRG according to Wright (2006). +To estimate the background and foreground contamination, we +create a HEALPix (Gorski et al. 2005; Zonca et al. 2019) map of +all galaxies in the EDR using pixel dimensions of 0.00082 deg2 +and define an annulus around each LRG between 0.4 and 0.5 degree +radius. We determine that, at this distance, deviations from large scale +structure start to dominate over the Poisson uncertainties. Because +a fixed angular scale will measure different background at different +redshifts, we test multiple apertures and find that our results are +insensitive to background aperture size. We identify the HEALPix +pixels within the background annulus and use the galaxies within +those pixels as our background sources. This is advantageous as +HEALPix allows us to very quickly index which galaxies fall in +which part of the sky. We do this separately for each LRG, which +allows our background estimate to trace the local large scale structure +on which each LRG lies. At the median redshift of each of our redshift +bins, 0.45 degrees, corresponds to 9.96 Mpc, well beyond the size of +the largest virialized clusters. +To minimize background and foreground contamination, we use +the UltraVISTA catalog make cuts in observed (𝑔 − 𝑟) and (𝑟 − 𝑧) +color for each redshift bin. We make cuts that maximize the number +of galaxies in our redshift range and minimize the galaxies outside +our redshift range. For example, we find that galaxies with (𝑟 − 𝑧) +< 0.9(𝑔 − 𝑟) and (𝑟 − 𝑧) < 1.3 are in the redshift range 0.35 < +z < 0.5. We reduce background and foreground contamination by +only considering galaxies within these bounds at these redshifts. For +redshift bins 0.2 < z < 0.35 the cuts are made at (𝑟 − 𝑧) = (𝑔 − 𝑟) and +(𝑟 − 𝑧) = 1.1 and for redshift bin 0.5 < z < 0.65 the cuts are made at +(𝑟 − 𝑧) = (𝑔 − 𝑟) + 0.2 and (𝑟 − 𝑧) = 1.6. An example of these cuts +can be seen in Fig. 4. Table 1 reports the percentage of target sources +retained by the cut, as well as the percentage of sources retained that +are actually outside the redshift slice. This method was most effective +at eliminating sources above our target redshift range. These cuts are +implemented at all stages of the analysis, and significantly reduce the +Poisson uncertainty in our background estimation. +For each LRG, we place near neighbors, background, and fore- +ground galaxies in their own color-color-magnitude diagram, using +their observed (𝑟 − 𝑧) and (𝑔 − 𝑟) colors and observed 𝑧-band mag- +nitude. The grid consists of 50x50x50 cells, spanning -1.8 < (𝑟 − 𝑧) +< 10.4, -6.5 < (𝑔 − 𝑟) < 10.6, and 13 < 𝑧-band magnitude < 23, +corresponding to the range of our values in our catalog. The number +of background galaxies N𝑏𝑘𝑔 in each color-color-magnitude cell is +then scaled to the angular area of the 600 kpc search radius to reflect +the number of background galaxies we would expect to see within +that area. +In performing this normalization, we account for the amount of +area of the sky lost due to bright stars and large foreground galaxies in +two ways. For the background, we identify Legacy Survey galaxies +that are flagged for touching a Tycho-2 or Gaia star (to a 𝑔-band +magnitude < 13), a large galaxy, or a star cluster. We determine +in which HEALPix pixels in the background annulus the flagged +sources are found and exclude the flagged pixel from our analysis. +The median percentage of our background area with compromised +photometry is less than two percent. +For near neighbors, we use a different method because the area +of our HEALPix pixels is too large and results in the elimination of +MNRAS 000, 1–14 (2022) + +6 +M. E. Townsend et al. +Table 1. Results of color-color cut for each redshift bin. The table reports the percentage of sources that are truly in the redshift bin that are retained by the cut, +as well as the percentage of interloper sources retained that are actually outside the redshift slice. This method was most effective at eliminating sources above +our target redshift range. +Results of color-color cut +redshift bin +percentage of true sources retained +percentage of low-z interlopers retained +percentage of high-z interlopers retained +0.2 < z < 0.35 +96.6% +81.6% +27.3% +0.35 < z < 0.5 +97.4% +91.8% +16.6% +0.5 < z < 0.65 +96.7% +98.8% +11% +1 +0 +1 +2 +3 +4 +(g-r) color +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(r-z) color +color cut boundary +z > 0.5 +z < 0.35 +0.35 < z < 0.5 +Figure 4. An example of the color-color cuts made for redshift bin 0.35 < z < +0.5 using galaxies from the UltraVISTA catalog. The cuts primarily eliminate +galaxies that are at higher redshift than the target redshift slice. +too much area within the 600 kpc search radius. Furthermore, we +find that the area close to our LRGs are not contaminated by bright +foreground galaxies or star clusters, so the only contamination is +from bright stars. Rather than using HEALPix, we use the k-d tree +algorithm to find Gaia DR2 stars within our 600 kpc search radius. +When a star is found within the search radius, the region affected +by the star’s halo is calculated by Equation 7 as prescribed by the +Legacy Survey pipeline★★ +𝑅𝐺 = 150 × 2.511−𝑔𝑚𝑎𝑔𝐺 × (0.262/3600) +(7) +where R𝐺 is the resulting radius in degree and gmag𝐺 is the +observed 𝑔-band magnitude for the star in the Gaia DR2 catalog. We +then use the radius to determine the area lost to contamination and +remove any flagged near neighbor sources from the analysis. The +median percentage of the search areas lost to contamination is less +than nine percent. Both of these contamination calculations are done +for each LRG. Near neighbors and the background are then scaled by +these area modifications. +Once these area corrections are made, we calculate the number +of satellites by subtracting the number of background galaxies from +the number of near neighbor galaxies, N𝑠𝑎𝑡,𝑖 = N𝑛𝑛,𝑖 - N𝑏𝑘𝑔,𝑖, on +a cell-by-cell basis in the color-color-magnitude diagram. We sum +over all color-color-magnitude bins to get a total number of satellite +galaxies for each system. Formally, this number can be negative due +★★ https://github.com/legacysurvey/legacypipe +to the variation of the density of the local background. However, these +measurements can be used to get a statistical estimate of the number +of satellite galaxies around LRGs at different redshifts. To measure +satellite numbers for our luminosity complete samples, we divide the +LRGs into three redshift bins: 0.2 < z < 0.35, 0.35 < z < 0.5, and 0.5 +< z < 0.65. We then apply selection matrices to each resulting color- +color-magnitude diagram to determine the number of satellites per +LRG above the corresponding luminosity limit. Selection matrices +are described in the next subsection. +3.2 +Satellite Identification in Bins of Luminosity and Redshift +In Sec. 2.4 we describe our luminosity completeness limits, derived +from the UltraVISTA catalog. One way to measure the rest-frame lu- +minosity of our satellite galaxies would be to assume that all galaxies +are at the LRG redshift and to then fit the sparsely sampled SEDs to +derive rest-frame luminosities. However, the vast majority of galax- +ies in close projection around any LRG are not at the LRG redshift +and this would cause significant systematic errors in the luminosity +(Rudnick et al. 2009). Instead, we decide to establish a color-color +dependent 𝑧-band magnitude limit, brighter than which galaxies at +the LRG redshift would be above our luminosity limit. We map these +luminosity completeness limits onto an observed 𝑧-band magnitude. +We perform this translation of rest-frame to observed properties, +as our lack of redshift information for satellite galaxies makes it +impossible to directly compute their rest-frame luminosities. To do +this, we first match the Legacy Survey catalog to UltraVISTA so +we can use Legacy Survey photometry and UltraVISTA redshifts. +Then, in each (𝑟 − 𝑧) and (𝑔 − 𝑟) cell we determine the median +observed 𝑧-band magnitude of UltraVISTA galaxies that are within +Δ log10(𝐿𝑔/𝐿⊙) = 0.2 of the luminosity limit and in bins of 0.05 +in redshift in each color-color cell. We do this for each combina- +tion of luminosity limit and redshift. This median 𝑧-band magnitude +in each color-color cell defines a selection boundary in color-color +space brighter than which galaxies at the redshift of the LRG are lu- +minosity complete. For example, if a color-color cell has a selection +limit of 𝑧 = 20 at the redshift of the LRG, then only cells brighter +than that limit will contribute to the total satellite count. This method +assumes that all galaxies around the LRG that are in excess above +the background are at the redshift of the LRG. The color dependence +of the 𝑧-band magnitude limit accounts for the SED shape variations +among satellite galaxies. +4 +Results +4.1 +Distribution of Measured 𝑁sat +We find the number distribution of satellite galaxies N𝑠𝑎𝑡 in each +redshift and luminosity bin. For LRGs at redshift 0.2 < z < 0.35 +we find the distribution for luminosity bins log10(𝐿𝑔/𝐿 ⊙) ≥ 9.27, +log10(𝐿𝑔/𝐿 ⊙) ≥ 9.58, and log10(𝐿𝑔/𝐿 ⊙) ≥ 9.85. For LRGs in the +redshift range of 0.35 < z < 0.5 we show the distribution for the +log10(𝐿𝑔/𝐿 ⊙) ≥ 9.58 and log10(𝐿𝑔/𝐿 ⊙) ≥ 9.85 bins. For LRGs in +MNRAS 000, 1–14 (2022) + +LRG satellites +7 +the redshift range 0.5 < z < 0.65 we show the distribution for the +log10(𝐿𝑔/𝐿 ⊙) ≥ 9.85 luminosity bin, only. These distributions can +be found in the appendix, and they represent the number distribution +of possible satellite galaxies above a certain luminosity threshold for +LRGs in the redshift range. +In all redshift and luminosity bins, there is a large scatter in the +distribution and a tail toward higher satellite numbers. An example of +our N𝑠𝑎𝑡 distributions can be found in the top panel of Fig. 5. Despite +the tail of 𝑁𝑠𝑎𝑡 to high values as seen, e.g. in the top of Fig. 5 and +Appendix A, the large spread in 𝑁𝑠𝑎𝑡 makes it difficult to determine +at face value how many LRGs have a statistically significant detection +of satellites. +To determine the significance of our N𝑠𝑎𝑡 measurement, we deter- +mine how many "satellites" we would expect to detect in a random +pointing. This constitutes the null prediction for our measurement. +For each luminosity bin, we randomly select 10,000 galaxies from +the EDR. We consider these randomly selected galaxies to be our +mock LRGs. We then randomly assign each mock LRG a redshift +between 0.2 < 𝑧 < 0.65. We then run our statistical background +subtraction method on these mock LRGs and apply the selection +matrices at the mock redshift. N𝑛𝑛,𝑛𝑢𝑙𝑙, N𝑏𝑘𝑔,𝑛𝑢𝑙𝑙 and N𝑠𝑎𝑡,𝑛𝑢𝑙𝑙 are +determined in the same way as N𝑛𝑛, N𝑏𝑘𝑔 and N𝑠𝑎𝑡 from the real +data. +An example of the distribution of satellites gleaned from the null +test is in the second panel from the top of Fig. 5 and the results of the +null test for all luminosity bins can be found in Figs. A1 through A6. +Results of the null tests for each luminosity-complete sample are used +to determine the significance of N𝑠𝑎𝑡 measurements. We consider +there to be a significant satellite detection if the measurement N𝑠𝑎𝑡 for +an individual LRG exceeds the 99th percentile of satellites measured +from the null result. The 99th percentile is noted on the plots of +satellite numbers throughout this paper. +In Fig. 6 we show the distribution of 𝑁𝑠𝑎𝑡 vs. redshift for all of +our combinations of redshift and luminosity limit. In each panel we +denote the 99th percentile for a significant 𝑁𝑠𝑎𝑡 detection as a hori- +zontal line. The 99th percentile is different for each combination of +redshift and luminosity. At fixed luminosity (row) the threshold for +a significant detection decreases with increasing redshift. While the +99% limit decreases towards higher redshift, the fraction of LRGs +with a significant number of satellites is constant within the uncer- +tainties, as shown in the third row of Table 2. This likely stems from +the different Poisson uncertainty from the background subtraction in +different redshift bins. On one hand, our constant luminosity thresh- +old will correspond to a brighter 𝑧-band magnitude limit at lower +redshift, which should decrease the number of interlopers. On the +other hand, our color cuts described in Sec. 3.1 are very effective at +removing high redshift (and likely fainter) interlopers, but reject the +lowest fraction of interlopers for our lowest redshift bins and would +result in a higher contribution to the uncertainty in 𝑁𝑠𝑎𝑡. The combi- +nation of these effects is likely responsible for the modest dependence +of the 99% confidence level on redshift. +Most LRGs in all bins do not have a significant satellite mea- +surement within a 600 kpc radius. Some LRGs have small negative +number of satellites. This is a result of the statistical nature of our sub- +traction method. The existence of negative satellites - an obviously +unphysical phenomenon - is a reflection of the random fluctuations +in the background measurement. At times, those fluctuations will +produce a high background when compared to near neighbors. +4.2 +Color Distribution of LRG Satellites +While our method makes it impossible to study individual satellite +galaxies in detail, it is possible to determine color and magnitude +40 +20 +0 +20 +40 +60 +80 +100 +0 +5 +10 +counts +mean = 8.6 +median = 6.014 +percentage above +99% level: 7.4% +Data +40 +20 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +counts +mean = 0.8689 +median = -0.4079 +KS pval = 1.217e-09 +Null +40 +20 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +counts +mean = 4.254 +median = 2.297 +KS pval = 0.0008567 +percentage above +99% level: 2.9% +Nsat, mod = 1 model +40 +20 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +counts +mean = 6.205 +median = 4.398 +KS pval = 0.2029 +percentage above +99% level: 3.4% +Nsat, mod = 3 model +40 +20 +0 +20 +40 +60 +80 +100 +Number of Satellites +0 +20 +40 +60 +80 +counts +mean = 9.937 +median = 8.218 +KS pval = 0.002221 +percentage above +99% level: 4.4% +Nsat, mod = 7 model +Nsat, mod Histograms for log10(Lg/L ) > 9.27 and 0.2 < z < 0.35 +Figure 5. Distributions of the number of satellites found from the data +(top panel), the null test (second panel), and models for different intrinsic +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑 ⟩ numbers for 0.2 < z < 0.35 and log10(𝐿𝑔/𝐿⊙) ≥ 9.27 (bottom +three panels). The dot-dash line indicates the 99th percent confidence for each +bin, as described in Sec. 4.1. Above this line, the number of satellites would +only occur less than 1 percent of the time in a randomly drawn sample of the +sky. The null test and the models are described in Sec. 4.1. ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 +and 7 illustrate models that are not consistent with the data, with too few and +with a median number of satellites that is too low and too high, respectively. +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑 ⟩ = 3 shows a model that is consistent with the data, as illustrated +with a p-value = 0.2. All panels show a skewed distribution, with a tail towards +high satellite counts. However, the data and models show a more pronounced +tail than the null prediction. The plots also indicate the percentage of LRGs +in both the data and the models that exceed the number of satellites at 99 +percent confidence. +distributions as a population. Fig. 7 illustrates the observed (𝑟 − 𝑧) +color vs observed 𝑧-band magnitude of satellites compared to LRGs +in bins of redshift and luminosity. We show the distribution of LRGs +and satellites for the full sample. +In general, the satellite population is fainter than the population of +LRGs, though there is some overlap in the distributions. In all cases +satellite galaxies appear to have (𝑟 − 𝑧) colors similar to or bluer than +that of LRGs. +To quantify the difference in observed (𝑟 − 𝑧) color between LRGs +and their satellites, we implement the following process. We cannot +identify an individual galaxy as a satellite, hence we do not know +the exact satellite (𝑟 − 𝑧) colors. As an approximation, we use the +midpoint value of the (𝑟 − 𝑧) color bins as satellite colors. Each +(𝑟 − 𝑧) color bin is weighted by the number of galaxies in it. We then +subtract this color from the color of the host LRG and we interpolate +the resulting Δ(𝑟 − 𝑧) onto a reference grid. +We show the distribution of Δ(𝑟 − 𝑧) in the left panel of Fig. 8 for +each redshift and luminosity bin. The right panel shows the medians +of the distributions for each luminosity and redshift bin with the error +MNRAS 000, 1–14 (2022) + +8 +M. E. Townsend et al. +0.20 +0.25 +0.30 +0.35 +40 +20 +0 +20 +40 +60 +80 +100 +Nsat +0.2 < z < 0.35 +log10(Lg/L +) + 9.27 +99 percent confidence = 61.73 +0.20 +0.25 +0.30 +0.35 +40 +20 +0 +20 +40 +60 +80 +100 +Nsat +log10(Lg/L +) + 9.58 +99 percent confidence = 53.76 +0.35 +0.40 +0.45 +0.50 +0.35 < z < 0.5 +99 percent confidence = 33.69 +0.20 +0.25 +0.30 +0.35 +redshift +40 +20 +0 +20 +40 +60 +80 +100 +Nsat +log10(Lg/L +) + 9.85 +99 percent confidence = 52.12 +0.35 +0.40 +0.45 +0.50 +redshift +99 percent confidence = 32.23 +0.50 +0.55 +0.60 +0.65 +redshift +0.5 < z < 0.65 +99 percent confidence = 30.46 +Figure 6. The number of satellites for each LRG plotted against the LRG redshift. The horizontal lines indicate the 99th percent confidence for each bin, as +described in Sec. 4.1. Above this line, the number of satellites would only occur less than 1 percent of the time. Fewer satellites are needed for a significant +detection in the high redshift bins compared to the lower redshift bins, at fixed luminosity. Table 2 gives the proportion of LRGs in each bin that have significant +satellite detections, as well as the upper and lower bounds given by a 99% binomial confidence interval. Despite the lower threshold, at fixed luminosity the +proportion of LRGs with significant detections decreases with increasing redshift. +bars representing the 68 percent confidence interval in the weighted +median determined by bootstrap resampling. The right panel in Fig. +8 shows no statistical evidence for redshift evolution. +4.3 +Forward Modeling the Satellite Population +With the above techniques we can infer the number of LRGs with a +statistically significant excess of satellites. However, because of the +large spread in 𝑁𝑠𝑎𝑡 and other potential systematics (see previous +sections), we cannot straightforwardly determine the average number +of satellites per LRG using only the measured data. For this reason +we develop a method in which we use our null distribution of 𝑁𝑠𝑎𝑡 +to forward model the predicted distribution of the measured 𝑁𝑠𝑎𝑡 for +an intrinsic distribution of satellites. Modeling in this way attempts +to answer the question, if we integrate down the satellite galaxy +luminosity function to a certain luminosity and at a certain redshift, +would we find that the number distribution of satellite galaxies around +LRGs to be consistent with LRGs having an intrinsic number of +satellites? +We start with the assumption that the mock LRGs in a given model +have an intrinsic mean number of satellites, with the number around +each mock LRG independently drawn from a Poisson distribution +with the corresponding expectation value. This number is effectively +the integral of the luminosity function of satellites down to the lumi- +nosity threshold. For example, a model with an expectation value of +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 would have all mock LRGs populated with a number +of satellites drawn from a Poisson distribution with that expectation +value. +MNRAS 000, 1–14 (2022) + +LRG satellites +9 +0 +1 +2 +(r-z) color +0.2 < z < 0.35 +log10(Lg/L +) + 9.27 +LRG +0 +1 +2 +(r-z) color +log10(Lg/L +) + 9.58 +LRG +0.35 < z < 0.5 +LRG +16 +17 +18 +19 +20 +21 +22 +observed z-band magnitude +0 +1 +2 +(r-z) color +log10(Lg/L +) + 9.85 +LRG +16 +17 +18 +19 +20 +21 +22 +observed z-band magnitude +LRG +16 +17 +18 +19 +20 +21 +22 +observed z-band magnitude +0.5 < z < 0.65 +LRG +0 +20 +40 +60 +80 +100 +120 +140 +counts +0 +20 +40 +60 +80 +100 +120 +140 +counts +0 +20 +40 +60 +80 +100 +120 +140 +160 +counts +0 +20 +40 +60 +80 +100 +120 +140 +counts +0 +25 +50 +75 +100 +125 +150 +175 +counts +0 +20 +40 +60 +80 +100 +120 +140 +160 +counts +Figure 7. Observed (𝑟 − 𝑧) color vs. 𝑧-band magnitude diagram for LRGs (red dots) and satellite galaxies (shading) for different redshift and luminosity bins. +Each red dot represents an individual LRG in this redshift range and the shaded regions represent the distribution of satellite galaxies in bins of (𝑟 − 𝑧) observed +color and observed 𝑧-band magnitude. The vertical lines across the top of the first panel represents the median uncertainty in the color measurement at the +observed 𝑧-band magnitude at which they are placed. In general, the satellite population is fainter than the population of LRGs, though there is some overlap in +the distributions. In all cases satellite galaxies appear to have (𝑟 − 𝑧) colors similar to or bluer than that of LRGs. We quantify this in Fig. 8. +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + (r-z) +0 +250 +500 +750 +1000 +1250 +1500 +1750 +counts +0.2 < z < 0.35 +0.35 < z < 0.5 +0.5 < z < 0.65 +log10(Lg/L +) + 9.27 +log10(Lg/L +) + 9.58 +log10(Lg/L +) + 9.85 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +redshift +0.09 +0.10 +0.11 +0.12 +0.13 +0.14 +0.15 +0.16 +0.17 +median of (r-z) +Figure 8. Left: The distribution of the difference between the observed (r-z) colors of satellites and their host LRGs, (𝑟 − 𝑧)𝐿𝑅𝐺 - (𝑟 − 𝑧)𝑠𝑎𝑡 = Δ(𝑟 − 𝑧). Colors +represent different luminosity limits and line styles represent different redshift ranges. Right: The median value of Δ(𝑟 − 𝑧) for each luminosity and redshift bin. +Error bars represent 16th to 84th percentile in the weighted median determined through boostrap resampling. +MNRAS 000, 1–14 (2022) + +10 +M. E. Townsend et al. +0 +2 +4 +6 +8 +10 +12 +Nsat, mod +log10(Lg/L +) + 9.27 +log10(Lg/L +) + 9.58 +log10(Lg/L +) + 9.85 +Tal+2012 +SAMs +0 +2 +4 +6 +8 +10 +Nsat, mod +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +redshift +0 +2 +4 +6 +8 +10 +Nsat, mod +Figure 9. A summary of our forward modelling results, described in Sec. 4.3 and Sec. 5. The y-axis is the number of model satellites and the x-axis is redshift. +The shaded regions show the range of numbers of satellites with a p > 0.05 and the dashed boxes show the range with a p > 0.003. Each panel includes our +forward modeling results as rectangles and results from an analysis of the satellite counts from semi-analytic models of Hirschmann et al. (2016) as square +points. The error bars for the SAMs are the 68% confidence limits on the distribution; the errors in the mean are smaller than the points and are thus not plotted. +The bottom panel also includes the satellite numbers for two redshift bins from Tal et al. (2012), shown as round black points. These are scaled to our search +radius of 600 kpc and the error bars represent the 68% confidence limits determined by the range of 𝛼𝑠 in Tal et al. (2012). +MNRAS 000, 1–14 (2022) + +LRG satellites +11 +Table 2. Table summary of satellite count and forward modeling. N𝐿𝑅𝐺, N𝐿𝑅𝐺(sig. satellites), and N𝐿𝑅𝐺(sig. satellites)/N𝐿𝑅𝐺 are the total number of LRGs, +the total number of LRGs with a significant satellite detection, and the proportion of LRGs with significant satellite counts with binomial confidence intervals, +respectively. The row labeled “⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ range” are the range of values of ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ drawn from a Poisson distribution that are consistent with the data +to the 95% confidence (99.75% confidence). +Summary Table of Satellite Counts and Forward Modeling +log10(L𝑔/L⊙) ≥ 9.27 +log10(L𝑔/L⊙) ≥ 9.58 +log10(L𝑔/L⊙) ≥ 9.85 +0.2 ≤ z < 0.35 +0.2 ≤ z < 0.35 +0.35 ≤ z < 0.5 +0.2 ≤ z < 0.35 +0.35 ≤ z < 0.5 +0.5 ≤ z ≤ 0.65 +N𝐿𝑅𝐺 +309 +309 +617 +309 +617 +897 +N𝐿𝑅𝐺(sig. satellites) +14 +16 +16 +14 +15 +12 +N𝐿𝑅𝐺(sig. satellites)/N𝐿𝑅𝐺 +0.0450.085 +0.020 +0.0410.093 +0.025 +0.0260.047 +0.012 +0.0450.085 +0.020 +0.0240.045 +0.011 +0.0130.027 +0.006 +⟨ N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ range +6.0-10.2 (4.0-11.5) +5.0-10.0 (3.0-11.1) +1.7-3.0 (1.2-4.1) +4.0-10.0 (2.4-11.0) +2.0-3.0 (1.0-4.3) +1.0-2.4 (0.3-2.5) +After we determine how many satellites to add to each mock LRG, +we add sources to the near neighbor matrix N𝑛𝑛,𝑛𝑢𝑙𝑙 in cells that +are at least as bright in the median observed 𝑧-band magnitude for +the color-color cell. In practice, these galaxies are randomly chosen +duplicates of galaxies already in those cells, to ensure that the added +mock satellites follow the same color-color-magnitude distribution of +the actual galaxies. The resulting matrix is N𝑛𝑛,𝑚𝑜𝑑. The background +matrix for the mock LRG remains unmodified. The model number +of satellites, N𝑠𝑎𝑡,𝑚𝑜𝑑, is calculated by subtracting N𝑏𝑘𝑔,𝑛𝑢𝑙𝑙 from +N𝑛𝑛,𝑚𝑜𝑑, and then applying the appropriate selection matrix. +A sample of our model results is shown in Fig. 5, which indicates +the mean, median, and p-value from the Komogorov-Smirnov 2- +sample test. The p-value indicates whether or not the model and data +are consistent with being drawn from the same distribution. Small +p-values indicate that the model and the data likely did not come +from the same distribution. For example, the case shown in Fig. 5 for +0.2 < z < 0.35 and log10(𝐿𝑔/𝐿 ⊙) ≥ 9.27 has an acceptable p-value +(> 0.003) for the model with ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3, but significantly lower +p-values for all other ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩, including the null result of zero +mean satellites. The remaining comparisons between the data, null +result, and models can be found in Appendix A. +Fig. 9 shows a summary of our model results. The solid boxes +represent the range of satellite numbers expected for p > 0.05 and the +dashed boxes represent the range for p > 0.003. The colors represent +the same luminosity bins as Fig. 6. Based on these results, we can +rule out a lack of evolution to 𝑧 < 0.5 at a fixed luminosity limit at +the greater than 95% level. We conclude this as the allowed range +of ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ at 𝑧 < 0.5 in each luminosity bin do not overlap. We +cannot reject the no-evolution hypothesis at the greater than 99.7% +level as the allowed values at that confidence level overlap across all +redshifts. +We do not have the precision to constrain the faint end of the +luminosity function, but the lack of a clear increase in ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ +with decreasing luminosity limit in Fig. 9 implies a flat or declining +faint-end luminosity slope for satellites. A summary of these result +can be found in Table 2. +5 +Discussion +5.1 +Colors and Numbers of LRG satellites +While our statistical subtraction methods leaves us unable to study +any single satellite galaxy in particular, nevertheless we are able to +study satellite properties on a population level. +Fig. 8 illustrates the difference in the color distribution between +satellites and their host LRG. The color difference across all bins +are consistent with each other. The color distribution for the highest +redshift and highest luminosity bin has a significantly larger spread +than the other bins and extends to fainter luminosities. The origin of +this is seen in the bottom right-hand panel of Fig. 7 in which we see +that the LRGs and the reddest satellites are significantly redder than in +lower redshift bins, though the satellites still reach to as blue colors +as in the other redshift bins. This behavior stems from the color- +dependent k-corrections, i.e. the observed (𝑟 − 𝑧) color of passive +galaxies becomes rapidly redder with increasing redshift while the +observed color for blue galaxies changes much less. The result is an +increase in the difference between the reddest and bluest objects in +observed color as one looks to higher redshifts. +As shown in Fig. 9, at a fixed redshift we see substantial overlap +between acceptable models for each luminosity limit, such that we +cannot rule that the mean number of satellites is independent of the +luminosity limit, at least over our luminosity range. Despite that, +there are some hints of a dependence on the luminosity limit. In the +low redshift bin we can see that the 2− and 3 − 𝜎 lower bounds +of acceptable models increases as the luminosity limit decreases. +For log10(𝐿𝑔/𝐿 ⊙) ≥ 9.27, the lower bound is ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 4. The +lower 3 − 𝜎 bound for log10(𝐿𝑔/𝐿 ⊙) ≥ 9.58 is ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 and +for log10(𝐿𝑔/𝐿 ⊙) ≥ 9.85 ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 2.4. This is somewhat as +expected, as we should be increasing the number of satellites as we +integrate further down the luminosity function. +As described in Sec. 4.3, there is tentative evidence for evolution +at 𝑧 < 0.5 in satellites selected to a fixed luminosity. Interpreting +this evolution in terms of the true number of satellites down to a +given mass limit is complicat ed, as satellites will evolve in 𝑀/𝐿★ +over time and as such should on average be reducing their luminosity. +This implies that satellites which are just visible above our luminosity +limit at higher redshift may fade below our limit by a lower redshift. +The tentative evolution we see may therefore be indicative of an even +stronger evolution in the number of satellites to a fixed stellar mass +threshold. We will address this issue in a future paper. +We compare our results to Tal et al. (2012, hereafter T12) by +integrating the luminosity function they found for satellites. The T12 +analysis differs from ours in important ways. First, they only go +down to a luminosity of log10(𝐿𝑔/𝐿 ⊙) ∼ 10 at 0.28 < z < 0.4 and +log10(𝐿𝑔/𝐿 ⊙) ∼ 10.5 at 0.6 < z < 0.7, whereas this work goes down to +log10(𝐿𝑔/𝐿 ⊙ ∼ 9.85 out to 𝑧 = 0.65. Second, T12 measures satellites +out to 1 Mpc, which is a larger radius than we search in this work. To +account for these differences we first integrate the luminosity function +from T12 down to their luminosity limits. These limits are brighter +than ours but from a small sample at lower redshift T12 showed +that the luminosity function starts flattening below log10(𝐿𝑔/𝐿 ⊙ ∼ +10. We therefore do not think that the T12 satellite numbers when +integrated to our luminosity limits would significantly increase. We +then account for the different in radial search areas by scaling the +T12 result by the ratio of the mean N𝑠𝑎𝑡 at 0.6 Mpc and 1 Mpc from +our own data +MNRAS 000, 1–14 (2022) + +12 +M. E. Townsend et al. +The T12 points are plotted in Fig. 9. The error bars represent the +68% confidence limits determined by the range of 𝛼𝑠 in T12. The +T12 points are in broad agreement with our data though they appear +to have a slightly higher number of satellites. +Additionally, we also compare our results to semi-analytic models +based on (Hirschmann et al. 2016, private communication)††. We +measure the number of satellites around simulated LRGs out to a 3D +radius of 600 kpc in bins of redshift and 𝑔-band luminosity. We plot +these model predictions in Fig. 9. There is good agreement between +our results and the semi-analytic models. At a fixed redshift, the +SAM has the same trend with luminosity limit as in our data. How- +ever, they show a shallower trend with redshift at a fixed luminosity +limit than does our data. It is not clear if this difference lies in the +approximations we make in our comparison, e.g. the use of 3D radii +for the models, or if it reflects a true problem with the simulations. +We will treat this in more depth in a future work. +5.2 +The Future Mass Growth of LRGs +We next estimate the amount of mass that can be gained by each LRG +resulting from satellite accretion at 𝑧 < 0.65. We cannot measure +the mass of individual satellite galaxies, but can use their statistical +color and luminosity distribution relative to that of the host LRGs to +compute the average amount of mass in satellites. We start with the +Δ(𝑟 − 𝑧) and Δzmag, the average difference between the observed +𝑧-band magnitude for LRGs and satellites. We use EZgal (Mancone +& Gonzalez 2012) to output a model from Conroy et al. (2009) +with a Chabrier initial mass function (Chabrier 2003). We used an +exponentially declining SFH from this model to define a relationship +between observed (𝑟 − 𝑧) color and the ratio of the stellar mass (𝑀★) +to the observed 𝑧-band flux ( 𝑓𝑧). This was done at the mean observed +redshift 𝑧𝑜𝑏𝑠 of each of our LRG subsamples. In this model the +relation between color and log10(𝑀★/ 𝑓𝑧) represents a trend with age. +This relation is valid for all smooth SFHs and dust extinction moves +galaxies relatively parallel to this relation. Using the mean difference +in observed color between LRGs and their satellites Δ(𝑟 − 𝑧), we use +this relation to infer a difference in log10(𝑀★/ 𝑓𝑧) between LRGs +and satellites, which we call Δ log10(𝑀★/ 𝑓𝑧). From this difference +and the ratio of the mean 𝑧-band flux between LRGs and satellites +we can compute the ratio of the stellar mass in the two populations +as +𝑙𝑜𝑔10 +� 𝑀★,𝐿𝑅𝐺 +𝑀★,𝑠𝑎𝑡 +� += Δ𝑙𝑜𝑔10 +� 𝑀★ +𝑓𝑧 +� +− 𝑙𝑜𝑔10 +� +𝑓𝑧,𝑠𝑎𝑡 +𝑓𝑧, 𝐿𝑅𝐺 +� +. +(8) +We make a number of assumptions and simplifications in this cal- +culation. First, we assume all satellites have the same star formation +history or can be represented by a single star formation history. In +reality, the satellites with lower luminosity may have different star +formation histories but at the same time all smooth SFHs should lie +in the same plane of log10(𝑀★/ 𝑓𝑧) vs. (𝑟 − 𝑧)𝑜𝑏𝑠. Additionally, we +assume that the star formation histories of LRGs and satellites are +the same and we assume that the effects of dust are negligible on the +relation we use. +We find that log10 +� 𝑀★,𝐿𝑅𝐺 +𝑀★,𝑠𝑎𝑡 +� +ranges from 0.77 to 1.22, with the +highest mass in satellites occurring in our highest redshift and highest +luminosity limit bin. The lowest mass in satellites occurs in our +lowest redshift and lowest luminosity limit bin. At a given redshift, +log10 +� 𝑀★,𝐿𝑅𝐺 +𝑀★,𝑠𝑎𝑡 +� +is constant to within ≲ 0.1 dex for satellites with +different luminosity limits. Furthermore, at the same luminosity limit, +†† https://sites.google.com/inaf.it/gaea/?pli=1 +but over different redshifts, log10 +� 𝑀★,𝐿𝑅𝐺 +𝑀★,𝑠𝑎𝑡 +� +increases as redshift +increases. The increase in relative satellite mass is ∼ 0.2 dex between +the two lowest redshift bins for the log10(𝐿𝑔) > 9.58 limit and ∼ 0.3 +dex between the two redshift lowest bins for the log10(𝐿𝑔) > 9.85 +limit. There is no increase in relative satellite mass between the +middle and highest redshift bin for the log10(𝐿𝑔) > 9.85 limit. +There are two reasons why the relative mass in satellites might +increase to higher redshift. First, it could be that there are more satel- +lites at higher redshift and that the rate of consumption by the LRGs +overtakes the accumulation of new satellites. Second, it could be that +all satellites will move to lower mass-to-light ratio values toward +higher redshift because they are younger, so a fixed luminosity limit +will probe to a lower satellite stellar mass limit at higher redshift. We +cannot distinguish between these two possibilities without measur- +ing the rest-frame colors of satellites or their stellar mass. However, +given these caveats we can say that LRGs can at most grow by ∼ 15% +from 𝑧 = 0.575 to the present day and by ∼ 6% from 𝑧 = 0.275 to +the present day. This is consistent with the findings from Brown +et al. (2008), which determined that the mass of massive red galaxies +grows by 30% since z < 1. +6 +Summary and Conclusions +In this study, we use the deep photometry from the DESI Legacy +Imaging Surveys to characterize the satellite population around +SDSS-identified luminous red galaxies. SDSS LRGs represent a ho- +mogeneous population of massive red ellipticals, ideal for investigat- +ing the mechanisms by which the mass of massive ellipticals builds +up. We use the Legacy Survey photometry for SDSS DR14 LRGs +froom the LOWZ and CMASS samples. The deeper Legacy Survey +photometry allows us to carry out a characterization of LRG satellites +to fainter luminosities and higher redshifts than previously possible +for a large sample. We have 1,823 LRGs in our 25 square degree +sample area. +We perform a statistical background subtraction in (𝑔 − 𝑟), (𝑟 − 𝑧), +and 𝑧-band magnitude space. To determine what counts as a signif- +icant satellite detection, an LRG must have a number of satellites +above the 99th percentile of the null result in each luminosity bin. +We develop a technique to forward model the satellite distribution, +accounting for the survey selection function and systematics in our +background subtraction technique. This method deals with the lack +of a priori knowledge of the redshifts and luminosities of the satellite +galaxies. +The main results of this paper are as follows: +• LRGs are in general brighter than their satellites in the observed +𝑧-band, but satellite galaxies have similar or slightly bluer observed +(𝑟 − 𝑧) colors to LRGs. This is illustrated in Figs. 7 and 8. +• We show in Fig. 6 and in Table 2 that the proportion of LRGs +that have significant satellite detections decreases with increasing +redshift. This does not have a straightforward interpretation with +regards to the true population of satellites, as the threshold for sig- +nificance varies with redshift and luminosity. +• To better interpret the number of satellites around the mean +LRG in each bin we developed a forward modeling approach in +which we use mock samples to simulate the observed signature of +difference satellite populations, assuming that the total number of +satellites is drawn from a Poisson distribution with expectation value +⟨ N𝑠𝑎𝑡,𝑚𝑜𝑑⟩. For LRGs at all combinations of redshift and rest-frame +𝑔-band satellite luminosity limit, we can strongly rule out a lack of +satellites (Fig. 9). +• At fixed redshift, the 3-𝜎 lower bounds go toward lower accept- +able 𝑁𝑠𝑎𝑡,𝑚𝑜𝑑. There is, however, significant overlap in acceptable +MNRAS 000, 1–14 (2022) + +LRG satellites +13 +models. This implies a flat or declining faint-end luminosity slope +for satellites. +• In our two highest luminosity bins we find tentative evidence +for evolution in ⟨ N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ out to 𝑧 < 0.5 though cannot rule +out the no-evolution hypothesis as the greater than 99.7% level. The +interpretation of this evolution is complicated by the expected 𝑀/𝐿★ +evolution of satellite galaxies over this redshift range. +• We calculated that LRGs can at most grow by ∼ 15% from +𝑧 = 0.575 to the present day and by ∼ 6% from 𝑧 = 0.275 to the +present day. +This pilot program has demonstrated the ability of the DESI +Legacy Imaging Survey, coupled with LRG spectroscopy, to probe +the satellite properties of the most massive galaxies in the Universe. +Despite the larger area of the EDR, it is not practical in our sample +to investigate the trends of satellite galaxies as a function of LRG +properties such as stellar mass. In a future work we will apply these +techniques to the full DESI Legacy Imaging Survey, in order to ana- +lyze the full population of SDSS-identified LRGs and to investigate +the luminosity function of satellite galaxies and their dependence on +LRG property. +Acknowledgements +The authors would like to thank John Moustakas at Siena College for +calculating the masses of LRGs used in this project. We thank Arjun +Dey and Stephanie Juneau for useful discussions that helped to spawn +this project. The authors acknowledge the support of NSF AST- +1716690, AST-1517815, NASA ADAP-80NSSC19K0592, ADAP- +NNX17AF25G, NASA HST-AR-14310.001-A. They also acknowl- +edge the support of the International Space Sciences Institute in Bern, +which hosted discussions of this work as part of the “The Effect of +Dense Environments on Gas in Galaxies over 10 Billion Years of +Cosmic Time” and “COSWEB: The Cosmic Web and Galaxy Evolu- +tion” teams. M.T. acknowledges the support of a University Graduate +Fellowship and the Lowry Graduate Fellowship from the University +of Kansas. G.R. thanks the Lorentz Center in Leiden for support +via their workshop “Galaxy Evolution in the Cosmic Web. G.R, +acknowledges the support of an ESO visiting science fellowship. +The Legacy Surveys consist of three individual and complemen- +tary projects: the Dark Energy Camera Legacy Survey (DECaLS; +Proposal ID #2014B-0404; PIs: David Schlegel and Arjun Dey), the +Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID #2015A-0801; +PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Sur- +vey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS +and MzLS together include data obtained, respectively, at the Blanco +telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIR- +Lab; the Bok telescope, Steward Observatory, University of Arizona; +and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. +The Legacy Surveys project is honored to be permitted to conduct +astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with +particular significance to the Tohono O’odham Nation. +NOIRLab is operated by the Association of Universities for Re- +search in Astronomy (AURA) under a cooperative agreement with +the National Science Foundation. +This project used data obtained with the Dark Energy Camera +(DECam), which was constructed by the Dark Energy Survey (DES) +collaboration. Funding for the DES Projects has been provided by the +U.S. Department of Energy, the U.S. National Science Foundation, +the Ministry of Science and Education of Spain, the Science and +Technology Facilities Council of the United Kingdom, the Higher +Education Funding Council for England, the National Center for +Supercomputing Applications at the University of Illinois at Urbana- +Champaign, the Kavli Institute of Cosmological Physics at the Uni- +versity of Chicago, Center for Cosmology and Astro-Particle Physics +at the Ohio State University, the Mitchell Institute for Fundamental +Physics and Astronomy at Texas A&M University, Financiadora de +Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Fi- +nanciadora de Estudos e Projetos, Fundacao Carlos Chagas Filho +de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Na- +cional de Desenvolvimento Cientifico e Tecnologico and the Minis- +terio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungs- +gemeinschaft and the Collaborating Institutions in the Dark Energy +Survey. The Collaborating Institutions are Argonne National Labo- +ratory, the University of California at Santa Cruz, the University of +Cambridge, Centro de Investigaciones Energeticas, Medioambien- +tales y Tecnologicas-Madrid, the University of Chicago, University +College London, the DES-Brazil Consortium, the University of Ed- +inburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, +Fermi National Accelerator Laboratory, the University of Illinois at +Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), +the Institut de Fisica d’Altes Energies, Lawrence Berkeley National +Laboratory, the Ludwig Maximilians Universitat Munchen and the +associated Excellence Cluster Universe, the University of Michigan, +NSF’s NOIRLab, the University of Nottingham, the Ohio State Uni- +versity, the University of Pennsylvania, the University of Portsmouth, +SLAC National Accelerator Laboratory, Stanford University, the Uni- +versity of Sussex, and Texas A&M University. +BASS is a key project of the Telescope Access Program (TAP), +which has been funded by the National Astronomical Observatories +of China, the Chinese Academy of Sciences (the Strategic Prior- +ity Research Program “The Emergence of Cosmological Structures” +Grant #XDB09000000), and the Special Fund for Astronomy from +the Ministry of Finance. The BASS is also supported by the Exter- +nal Cooperation Program of Chinese Academy of Sciences (Grant +#114A11KYSB20160057), and Chinese National Natural Science +Foundation (Grant #11433005). +The Legacy Survey team makes use of data products from the +Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), +which is a project of the Jet Propulsion Laboratory/California Insti- +tute of Technology. NEOWISE is funded by the National Aeronautics +and Space Administration. +The Legacy Surveys imaging of the DESI footprint is supported +by the Director, Office of Science, Office of High Energy Physics +of the U.S. Department of Energy under Contract No. DE-AC02- +05CH1123, by the National Energy Research Scientific Comput- +ing Center, a DOE Office of Science User Facility under the same +contract; and by the U.S. National Science Foundation, Division of +Astronomical Sciences under Contract No. AST-0950945 to NOAO. +Some of the results in this paper have been derived using the healpy +and HEALPix packages. +Data Availability +The data analyzed in this paper come from DR8 of the DESI +Legacy Imaging Survey. These data are available to the public +at https://www.legacysurvey.org/dr8/. Spectroscopic data is from +DR14 of the Slone Digitial Sky Survey and can be accessed at +https://www.sdss.org/dr14/spectro/spectro_access/. +REFERENCES +Banerji M., Ferreras I., Abdalla F. 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Lett., 373, 36 +White M., Zheng Z., Brown M. J. I., Dey A., Jannuzi B. T., 2007, Astrophys. +J., 655, L69 +Willmer C. N. A., et al., 2006, Astrophys. Journal, Vol. 647, Issue 2, pp. +853-873., 647, 853 +Wright E. L., 2006, PASP, 118, 1711 +Zonca A., Singer L., Lenz D., Reinecke M., Rosset C., Hivon E., Gorski K., +2019, Journal of Open Source Software, 4, 1298 +van Dokkum P. G., et al., 2008, Astrophys. J., 677, L5 +van Dokkum P. G., et al., 2010, Astrophys. J., 709, 1018 +van de Sande J., et al., 2013, Astrophys. J., 771 +van der Wel A., Holden B. P., Zirm A. W., Franx M., Rettura A., Illingworth +G. D., Ford H. C., 2008, Astrophys. J., 688, 48 +van der Wel A., et al., 2014, Astrophys. J., 788 +A +Satellite Distribution Modeling +Here we show the results of our LRG satellite models based on a +Poisson distribution in Figs. A1 through A6. Models that fit the data +with a p-value of 0.05 are shown, as well as examples of models that +were not good fits. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–14 (2022) + +LRG satellites +15 +50 +0 +50 +100 +150 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +counts +mean = 9.894 +median = 6.12 +percentage above +99% level: 4.5% +Data +50 +0 +50 +100 +150 +0 +20 +40 +60 +80 +counts +mean = 0.25 +median = -1.182 +KS pval = 3.121e-08 +Null +50 +0 +50 +100 +150 +0 +20 +40 +60 +80 +counts +mean = 2.221 +median = 0.782 +KS pval = 6.878e-05 +percentage above +99% level: 1.2% +Nsat, mod = 2 model +50 +0 +50 +100 +150 +Number of Satellites +0 +20 +40 +60 +80 +counts +mean = 7.768 +median = 5.997 +KS pval = 0.5436 +percentage above +99% level: 1.7% +Nsat, mod = 8 model +Nsat, mod Histograms for log10(Lg/L ) > 9.27 and 0.2 < z < 0.35 +Figure A1. Distributions of the number of satellites found from the data, the null test, and examples of Poisson models for 0.2 < z < 0.35 and log10(L) > 9.27. +A description of how models are formulated can be found in the text. The dot-dash line indicates 99 percent confidence. The distribution the data is compared +to the null test and the models. The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 8 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S +test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 2 is included as an example of model that does not have a good fit to the data. +MNRAS 000, 1–14 (2022) + +16 +M. E. Townsend et al. +50 +25 +0 +25 +50 +75 +100 +125 +0 +5 +10 +15 +counts +mean = 6.656 +median = 2.842 +percentage above +99% level: 5.2% +Data +50 +25 +0 +25 +50 +75 +100 +125 +0 +20 +40 +60 +80 +counts +mean = -2.051 +median = -3.576 +KS pval = 7.054e-07 +Null +50 +25 +0 +25 +50 +75 +100 +125 +0 +20 +40 +60 +80 +counts +mean = -1.099 +median = -2.546 +KS pval = 3.234e-05 +percentage above +99% level: 1.2% +Nsat, mod = 1 model +50 +25 +0 +25 +50 +75 +100 +125 +Number of Satellites +0 +20 +40 +60 +80 +counts +mean = 3.593 +median = 2.095 +KS pval = 0.436 +percentage above +99% level: 1.8% +Nsat, mod = 6 model +Nsat, mod Histograms for log10(Lg/L ) > 9.58 and 0.2 < z < 0.35 +Figure A2. Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.2 < z < 0.35 and log10(L) > 9.58. A description +of how models are formulated can be found in the text. The dot-dash line indicates the 99 percent confidence. The distribution the data is compared to the null +test and the models. The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 6 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 is included as an example of model that does not have a good fit to the data. +MNRAS 000, 1–14 (2022) + +LRG satellites +17 +20 +0 +20 +40 +60 +80 +100 +0 +5 +10 +15 +20 +25 +counts +mean = 4.54 +median = 2.754 +percentage above +99% level: 2.6% +Data +20 +0 +20 +40 +60 +80 +100 +0 +50 +100 +150 +counts +mean = 1.301 +median = 0.07083 +KS pval = 7.362e-07 +Null +20 +0 +20 +40 +60 +80 +100 +0 +50 +100 +150 +counts +mean = 4.197 +median = 3.051 +KS pval = 0.4182 +percentage above +99% level: 1.4% +Nsat, mod = 3 model +20 +0 +20 +40 +60 +80 +100 +Number of Satellites +0 +50 +100 +150 +counts +mean = 6.043 +median = 4.864 +KS pval = 3.045e-05 +percentage above +99% level: 1.6% +Nsat, mod = 5 model +Nsat, mod Histograms for log10(Lg/L ) > 9.58 and 0.35 < z < 0.5 +Figure A3. Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.35 < z < 0.5 and log10(L) > 9.58. A description +of how models are formulated can be found in the text. The dot-dash line indicates the 99 percent confidence. The distribution the data is compared to the null +test and the models. The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 5 is included as an example of model that does not have a good fit to the data. +MNRAS 000, 1–14 (2022) + +18 +M. E. Townsend et al. +50 +25 +0 +25 +50 +75 +100 +125 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +counts +mean = 5.311 +median = 2.639 +percentage above +99% level: 4.5% +Data +50 +25 +0 +25 +50 +75 +100 +125 +0 +20 +40 +60 +80 +counts +mean = -2.798 +median = -4.296 +KS pval = 7.855e-07 +Null +50 +25 +0 +25 +50 +75 +100 +125 +0 +20 +40 +60 +80 +counts +mean = -1.806 +median = -3.197 +KS pval = 5.633e-05 +percentage above +99% level: 1.2% +Nsat, mod = 1 model +50 +25 +0 +25 +50 +75 +100 +125 +Number of Satellites +0 +20 +40 +60 +80 +counts +mean = 3.852 +median = 2.577 +KS pval = 0.7454 +percentage above +99% level: 1.7% +Nsat, mod = 7 model +Nsat, mod Histograms for log10(Lg/L ) > 9.85 and 0.2 < z < 0.35 +Figure A4. Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.2 < z < 0.35 and log10(L) > 9.85. A description +of how models are formulated can be found in the text. The dot-dash line indicates the 99 percent confidence. The distribution the data is compared to the null +test and the models. The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 7 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 is included as an example of model that does not have a good fit to the data. +MNRAS 000, 1–14 (2022) + +LRG satellites +19 +20 +0 +20 +40 +60 +80 +0 +5 +10 +15 +20 +25 +counts +mean = 3.615 +median = 1.601 +percentage above +99% level: 2.4% +Data +20 +0 +20 +40 +60 +80 +0 +50 +100 +150 +counts +mean = 0.2848 +median = -0.7713 +KS pval = 5.761e-06 +Null +20 +0 +20 +40 +60 +80 +0 +50 +100 +150 +counts +mean = 3.115 +median = 1.951 +KS pval = 0.7539 +percentage above +99% level: 1.4% +Nsat, mod = 3 model +20 +0 +20 +40 +60 +80 +Number of Satellites +0 +50 +100 +150 +counts +mean = 5.068 +median = 4.003 +KS pval = 3.047e-05 +percentage above +99% level: 1.9% +Nsat, mod = 5 model +Nsat, mod Histograms for log10(Lg/L ) > 9.85 and 0.35 < z < 0.5 +Figure A5. Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.35 < z < 0.5 and log10(L) > 9.85. A description +of how models are formulated can be found in the text. The dot-dash line indicates the 99 percent confidence. The distribution the data is compared to the null +test and the models. The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 5 is included as an example of model that does not have a good fit to the data. +MNRAS 000, 1–14 (2022) + +20 +M. E. Townsend et al. +20 +0 +20 +40 +60 +0 +10 +20 +30 +counts +mean = 5.183 +median = 3.615 +percentage above +99% level: 1.3% +Data +20 +0 +20 +40 +60 +0 +50 +100 +150 +200 +counts +mean = 1.671 +median = 0.7421 +KS pval = 1.687e-18 +Null +20 +0 +20 +40 +60 +0 +50 +100 +150 +200 +counts +mean = 5.084 +median = 4.155 +KS pval = 0.4381 +percentage above +99% level: 1.3% +Nsat, mod = 2 model +20 +0 +20 +40 +60 +Number of Satellites +0 +50 +100 +150 +200 +counts +mean = 6.073 +median = 5.183 +KS pval = 9.111e-05 +percentage above +99% level: 1.5% +Nsat, mod = 3 model +Nsat, mod Histograms for log10(Lg/L ) > 9.85 and 0.5 < z < 0.65 +Figure A6. Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.5 < z < 0.65 and log10(L) > 9.85. A description +of how models are formulated can be found in the text. The dot-dash line indicates the 99 percent confidence. The distribution the data is compared to the null +test and the models. The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 2 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and +⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 is included as an example of model that does not have a good fit to the data. +MNRAS 000, 1–14 (2022) + diff --git a/h9E4T4oBgHgl3EQfrw0X/content/tmp_files/load_file.txt b/h9E4T4oBgHgl3EQfrw0X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96d69a20eff5fec0afefdcf3e376328ddefdbadd --- /dev/null +++ b/h9E4T4oBgHgl3EQfrw0X/content/tmp_files/load_file.txt @@ -0,0 +1,1643 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf,len=1642 +page_content='MNRAS 000, 1–14 (2022) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 The satellite population around luminous red galaxies in the 25 𝑑𝑒𝑔2 DESI Legacy Imaging Surveys Early Data Release Melinda Townsend,1★ and Gregory Rudnick1† 1University of Kansas, Department of Physics and Astronomy, 1082 Malott,1251 Wescoe Hall Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Lawrence, KS 66045 Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Luminous Red Galaxies, or LRGs, are representative of the most massive galaxies and were originally selected in the Sloan Digital Sky Survey as good tracers of large scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They are dominated by by uniformly old stellar populations, have low star formation rates, early type morphologies, and little cold gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Despite having old stellar populations and little in situ star formation, studies have shown that they have grown their stellar mass since z=1, implying that they grow predominantly via the accretion of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Tests of this picture have been limited because of the lack of deep imaging data sets that both covers a large enough area of the sky to contain substantial numbers of LRGs and that also is deep enough to detect faint satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use the 25 deg2 Early Data Release (EDR) of the DESI Legacy Imaging Surveys to characterize the satellite galaxy population of LRGs out to z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The DESI Legacy Imaging Surveys are comprised of 𝑔𝑟𝑧 imaging to 2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 mag deeper than SDSS and with better image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use a new statistical background technique to identify excess populations of putative satellite galaxies around 1823 LRGs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In three redshift and luminosity bins we measure the numbers of satellite galaxies and their 𝑟 − 𝑧 color distribution down to rest-frame 𝑔-band luminosity limits at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 times fainter than 𝐿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In addition, we develop a forward modeling technique and apply it to constrain the mean number of satellites in each of our redshift and luminosity bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Finally, we use these estimates to determine the amount of stellar mass growth in LRGs down to the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Key words: galaxies: evolution – galaxies: elliptical and lenticular, cD – Galaxy: general 1 Introduction The method by which massive galaxies accumulate their mass is an open question in galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Early-type galaxies (ETGs) have low specific star formation rates, old stellar populations, and very little cold gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Moustakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A homogeneous pop- ulation of early-type galaxies are Luminous Red Galaxies (LRGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Originally selected as good tracers of large scale structure (Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2001), subsequent investigations have shown them to be among the most massive galaxies in the universe, dominated by uniformly old stellar populations (Tojeiro & Percival 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Despite this, the stellar mass of LRGs has grown by 50% since z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='9, implying that they grow mostly through the accretion of passive satellites (Cool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Multiple studies have also shown that the spectra of LRGs indicate that those galaxies have undergone passive fading (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (Cool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Banerji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, some stud- ies find that only bright LRGs evolve passively while fainter LRGs have more extended star formation histories (SFG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Tojeiro & Percival 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' It is now well accepted that massive galaxies grow hierarchically through the successive effect of many mergers coupled with in situ growth by star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Frenk & White 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There is over- whelming observational support for this overall picture, with some of ★ E-mail: mtownsend@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='edu † E-mail: grudnick@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='edu the most direct evidence coming from the observations that massive galaxies at 𝑧 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8 have significantly smaller sizes and than similarly massive galaxies in the local universe (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Buitrago et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' van Dokkum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' van Dokkum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For example, Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2006) found that massive galaxies at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 < 𝑧 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7 were at least a factor of four times smaller in the rest- frame 𝑉-band than local counterparts of the same stellar mass, and that the the stellar density of these objects are at least 60 times larger than present-day massive ellipticals, indicating that they must have grown predominantly through dry mergers and Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2005) hypothesized that the size growth came from the accretion of satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Likewise, Man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2016) found that, to explain the ob- served number density evolution of massive galaxies, minor mergers are a necessary component to bring compact quiescent ellipticals into agreement with the stellar mass-size relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Accretion-based growth of passive galaxies leads to "inside-out" growth, where mas- sive galaxies grow via a series of minor mergers and a build-up of extended stellar halos (van Dokkum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Using luminosity functions of LRGs and their satellites, Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2012) determined that the mass ratio of LRGs to satellites is 4:1, making mass growth through major mergers unlikley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This is consistent with the findings in Bezanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2009), which find that central stellar densities of high redshift ellipticals are only a factor of a few higher when com- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='05210v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='GA] 12 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' pared to local ellipticals even though the total stellar density of high redshift ellipticals are orders of magnitude higher than their local counterparts, indicating that mass is being deposited in the outskirts of the galaxy through minor mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' van de Sande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2013) found that for fixed dynamical mass of massive quiescent galaxies from z ∼ 2, the mass density within one effective radius decreases by a factor of 20 while within a fixed physical radius of 1 kpc the mass density decreases by a factor of ∼ 2, which is consistent with inside-out growth through minor mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' These observational results are supported by simulations of galaxy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Naab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2010), using a high-resolution hydrodynamical cosmological simulation, shows that the accretion of weakly bound material - or minor mergers - can cause the radius of a massive spheroidal galaxy to increase as the square of the mass, whereas major mergers would cause the radius to increase at a linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This result echos the observations and implies a cause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' namely, that early-type galaxies are more compact at earlier times and grow less compact at low redshift because of the accretion of low-mass satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The rest-frame 𝐵-band luminsosity function of red galaxies also points to mergers as a mechanism to build up mass in massive sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2004) presented the rest-frame colors and lumi- nosities of ∼ 25,000 galaxies in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='78 deg2 Classifying Objects by Medium-Band Observations in 17 Filters (COMBO-17) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They found that the 𝐵-band luminosity den- sity does not evolve significantly in this redshift range, which, when coupled with the passive fading of the galaxy stellar populations, implies that there has been a build-up of stellar mass in the non-star forming population by a factor of 2 since z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Faber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2007) came to a similar result in their study that compares the luminosity functions from Willmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2006) of red and blue galaxies out to z ∼ 1 of the DEEP2 and COMBO-17 surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They also found a virtually constant 𝐵-band luminosity density for red galaxies since z ∼ 1, while the luminosity density of blue galaxies falls by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They argue that dry mergers are involved in building up present day massive ellipticals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Similarly, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2007) used the luminos- ity density and number density of galaxies to find that the amount of stellar mass contained in 𝐿∗ red galaxies has doubled since z = 1, but that the stellar mass in 4𝐿∗ red galaxies evolved more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In contrast, De Propris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2010) derived an upper limit for a dry merger rate in their measurement of the fraction of LRGs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='45 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 in dynamically close pairs taken from the 2dF-SDSS LRG and QSO (2SLAQ) redshift survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They find that minor dry mergers (a luminosity ratio 1:4 or higher) are unimportant to the mass build-up of the red sequence at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This holds at higher redshift, as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' mergers are found to not be the dominant channel for stellar mass build-up in early-type galaxies out to z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8 (Cimatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2006) and for quiescent galaxies out to z < 1 (Moustakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Scarlata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2007) determined that dry mergers are likely not a significant contributor to the build-up of massive ETGs, and that the most massive of these galaxies were already assembled 8 Gyr ago (although fainter ETGs keep assembling mass from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7 to the present (Tojeiro & Percival 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Banerji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2010) finds that the stellar mass function for LRGs with 𝑀★ > 3 × 1011M⊙ shows little evolution between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8, suggesting that most massive systems were in place by z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8 (see also Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There are theoretical reasons to believe that understanding accretion-based growth of massive ellipticals are important to un- derstanding how galaxies evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' De Lucia & Blaizot (2007) found that of z = 0 brightest cluster galaxies (BCGs) in the Millennium Simulation, only 10 percent were formed before z ∼ 1, and half were assembled after z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They also find that 50 percent of the stars found in BCGs were already formed by z ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Old stellar populations coupled with the late assembly times of these galaxies indicate that the BCGs in their sample gained most of their mass through accre- tion of satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Using mock catalogs constructed from the halo occupation distribution framework and comparing to data from the Boötes field for galaxies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='9, White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2007) found that their models overpredict the number of satellites that populate massive halos and interpret this as evidence that mas- sive satellite galaxies are merging or otherwise being disrupted by the central galaxy in that redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Building off this study, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2008) measures the halo occupation distribution for red galax- ies in the Boötes field and found that, while most massive galaxy stellar mass growth occurs prior to z = 1, massive galaxy growth continues and that a typical central galaxy grows by 30 percent from z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Although there is not uniform agreement, there are many indi- cations in both observations and simulations that merger events are responsible for the mass build up of massive ellipticals, but how this growth depends on redshift and primary galaxy mass is inconclu- sive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In addition, it is unclear what role is played by faint satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Most previous studies only probe satellites that are relatively mas- sive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' De Propris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2010) and Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2006) focused on the mass build up of early-type galaxies through major mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2009) probed satellites out to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 to log(M∗/M⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8, but for a combined field of less than a degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' It has been difficult to study faint satellites because of the dearth of data that can identify those faint satellites while at the same time encompassing a large sample of giant ellipticals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This situation is changing with the arrival of the DESI Legacy Imaging Surveys (Legacy Survey) (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' While large surveys have rev- olutionized the field of galaxy evolution, there has not until recently been data sets that both cover a large enough area of the sky to con- tain a substantial number of luminous ellipticals and deep enough to detect faint satellites out to intermediate redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Surveys like SDSS are too shallow to see very faint objects around luminous galaxies, and surveys like the NOAO Deep Wide-Field Survey only cover a small fraction of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In this paper, we combine the deep photom- etry of the Legacy Survey and the spectroscopic data from SDSS to characterize the satellite population around SDSS-identified LRGs 2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 magnitudes deeper than SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' By combining these two surveys, we solve the problem of shallow survey depth and limited sky coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use the two surveys in the following way: We use SDSS spectroscopy to select LRGs and the Legacy Survey imaging to detect faint candidate satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use statistical back- ground techniques to isolate likely satellites and study the abundance and properties of those satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This paper is structured as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2 describes the data and sample selection, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3 describes our analysis method, we present our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4, and discuss these results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 6 we present our summary and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We assume H0 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6, Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='286, and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='714 Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2 Data Description In this study we use SDSS-identified Luminous Red Galaxies using spectroscopic redshifts of SDSS and photometry from the DESI Legacy Imaging Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 SDSS Luminous Red Galaxies The sample of LRGs were identified in the Baryon Oscillation Spec- troscopic Survey (BOSS), part of the SDSS-III project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The BOSS LRGs are divided into a low redshift sample (LOWZ) and a constant- mass sample (CMASS) (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Both the LOWZ and MNRAS 000, 1–14 (2022) LRG satellites 3 240 241 242 243 244 245 246 247 RA (deg) 6 7 8 9 10 11 12 Dec (deg) 0 50 100 150 200 250 counts Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Density maps of the EDR after we eliminate Legacy Survey sources with observed 𝑧-band magnitude ≥ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The density variations reflect true variations in the large scale structure of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' CMASS samples were selected based on a set of color-magnitude cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For the LOWZ sample, these cuts are designed to only select the brightest and reddest galaxies at low redshift (z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4) and are similar to the SDSS- I/II Cut-I Luminous Red Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The LOWZ sample is also approximately volume-limited over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 and has a constant space density of ∼3 × 10−4 h3Mpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The CMASS sample includes galaxies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7 with an approximately con- stant stellar mass limit over the redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The CMASS sample is also selected by a color cut, similar to SDSS-I/II Cut-II and 2SLAQ LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, the cuts are bluer and more faint to increase the number density of targets in the CMASS redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Higher red- shift galaxies in CMASS are isolated using (𝑔 − 𝑟) and (𝑟 −𝑖) colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Together, LOWZ and CMASS make a spectroscopic sample that is 80 percent complete at log10(M/M⊙) ≥ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We choose to study LRGs out to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There are 151 LRGs between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='61 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 and may bias our sample in favor of older LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, we ran our analysis on samples with the maximum redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='61 and found no difference in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For more details on the selection criteria for LOWZ and CMASS (see Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Spectroscopic redshifts for this study are from SDSS DR14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We identify 1,823 LRGs in the EDR in the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We tested to see if LRG-LRG pairs skew our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We explicitly removed all LRGs that had an LRG within the approximate virial radius in redshift slices of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Of the 1,823 LRGs in this sample, 92 were removed because they were part of an LRG-LRG pair, leaving us with a sample of 1,731 LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The original 1,823 is our “pair” sample, because it includes LRG-LRG pairs, and the 1,731 LRGs is our “no pair” sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We calculated the distribution of satellite galaxies for both the “pair” and “no pair” sample and found that they were consistent with each other to within one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We conclude, then, that including LRG-LRG pairs does not alter our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 Legacy Survey Description The DESI Legacy Imaging Surveys‡ are a trio of surveys that im- age ≈ 14,000 deg2 of the extragalactic sky visible from the northern hemisphere in three optical bands - 𝑔, 𝑟, and 𝑧 - with a combined sur- vey footprint that is split into two contiguous regions by the galactic plane (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Those surveys are the Dark Energy Cam- era Legacy Survey, the Beijing-Arizona Sky Survey, and the Mayall 𝑧-band Legacy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The data used in this paper comes from the Dark Energy Camera Legacy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The Dark Energy Camera Legacy Survey (DECaLS) utilizes the Blanco 4-m telescope at Cerro Tololo Inter-American Observatory using the Dark Energy Camera and covers 9,000 deg2 in both the Northern and Southern Galactic Cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The survey obtains optical imaging data in the 𝑔-, 𝑟-, and 𝑧-bands, overlaps existing spec- troscopy from the Sloan Digital Sky Survey (SDSS), and reaches 5𝜎 point source depths of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4, and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 in the 𝑔-, 𝑟-, and 𝑧-band, respectively, more than two magnitudes deeper than SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Source detection is performed using The Tractor§, a forward mod- eling algorithm in which each source is modeled at the pixel level through simultaneous fits to a set of individual images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Each source is modeled using a set of parametric light profiles: a delta function, deVaucouleurs r−1/4 law, exponential disk, or exponential disk plus deVaucouleurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' See Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2018) for a more detailed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For this study we use the photometry from the 8th data release¶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For this study we use data from the area covered by the Legacy Survey Early Data Release (EDR), which covers between 240 and 245 degrees in right ascension and between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 degrees in declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 1 illustrates the z-band source density in the field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We want to establish a uniform detection limit across the survey area to make sure we are equally sensitive to satellites for all LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To do this, we use the 5𝜎 galaxy detection limit in the z-band to find the 90 percent brightest galaxy depth, and we limit our sample to galaxies brighter than this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We find this limit by binning sources into equal-size pixels using HEALPix∥ (Gorski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2019) and finding the median depth in each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' HEALPix discretizes the surface of a sphere into equal-area, non-overlapping tiles called pixels and astrophysical analysis can be done on a per-pixel basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Then we determine the 10th percentile and use that as our magnitude limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We find this limit to be a 𝑧-band magnitude 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='75 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We also confirm that all sources detected in the 𝑧-band are also detected in the 𝑔- and 𝑟-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The results of this 𝑧-band magnitude cut are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The density map for our magnitude complete sample demonstrates the presence of true large scale structure variations in our field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' As we will describe in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1, these variations motivate our use of a local background subtraction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 Rest Frame Magnitudes and Masses for LRGs To further characterize the SDSS LRGs, we compute both the rest frame 𝑟-band magnitudes and the stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We compute the rest-frame magnitudes for BOSS LRGs using the photometric red- shift code EAZY (Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008), fixing each LRG at its spectroscopic redshift and using as inputs the 𝑔𝑟𝑧 photometry from the Legacy Surveys, as well as WISE bands 𝑊1, 𝑊2, 𝑊3, and 𝑊4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' LRG stellar masses are computed using iSEDfit from Moustakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2013) assuming a Kroupa IMF (Kroupa 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' iSEDfit uses ‡ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='legacysurvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='org/ § http://thetractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='org/ ¶ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='legacysurvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='org/dr8/ ∥ http://healpix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='sourceforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='net MNRAS 000, 1–14 (2022) 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 redshift 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 log10(M*) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 redshift 23 22 21 20 19 Mr LOWZ CMASS Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The plot shows the log10(M★) and absolute 𝑟-band magnitude of LRGs in our sample v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' LRG redshift in the left- and right-hand panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' LOWZ LRGs are indicated by indigo squares and CMASS LRGs are indicated by green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Vertical lines show the bounds of our redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Baysian inference to extract physical properties from a galaxy’s ob- served broadband spectral energy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2 shows the stellar mass and r-band absolute magnitude as a function of redshift for SDSS LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The stellar mass (left plot) remains constant with redshift, although there is some scatter into lower stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Absolute 𝑟-band magnitude also shows scatter at low redshift toward fainter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We have 516 LRGs from the LOWZ sample and 1,307 from CMASS, for a total sample of 1,823 LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Of these, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7 percent have stellar masses below 1011 with 19 appearing in the LOWZ sample and 49 appearing in the CMASS sample, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 percent have M𝑟 > -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 with 19 appearing in the LOWZ sample and 41 appearing in the CMASS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Visual in- spection of these outliers show that these galaxies are predominantly red elliptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 Luminosity Completeness The redshift range of our LRG sample is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We wish to be equally complete to satellites above a given luminosity limit over a range in redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We decide to split our sample into three luminosity-complete subsamples, each spanning a different range in redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We determine the corresponding luminosity limit for each subsample using the UltraVISTA catalog from Muzzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2013), which is an ultra-deep 𝐾𝑠-selected catalog in the COSMOS field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The catalog covers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='62 deg2 and has photometry from 30 bands, to much deeper limits than the Legacy Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use this catalog to determine the 90 percent completeness limit in the rest-frame 𝑔-band luminosity for three redshift bins: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' As we wish to determine our luminosity in the DECam 𝑔-band filter, we use EAZY to synthesize 𝑔-band rest-frame luminosities using the DECam filter curve (Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We note that we do not use the 𝑟-band because our satellites are too faint to be detected in WISE 𝑊1 and 𝑊2 and therefore the 𝑟-band photometry would be unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' LRGs are bright enough to be detected in WISE, so we use the 𝑟-band luminosity for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Using only the UltraVISTA galaxies within our redshift range, we measure the luminosity down to which we recover 90 percent of the sources brighter than that limit at the high end of each red- shift bin, subject to our observed 𝑧-band magnitude limit of ≤ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We go through this process for each redshift bin and for all Ul- traVISTA sources, red UltraVISTA sources, and blue UltraVISTA sources (splitting the red and blue populations at (𝑈 − 𝑉) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We chose the most conservative luminosity where 90 percent of the galaxies were recovered for each redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3 shows the log10(𝐿𝑔/𝐿 ⊙) of the 𝑧-band magnitude ≤ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='75 UltraVISTA galaxies out to z = 1, with the solid vertical lines delini- ating the boundaries of our redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The three horizontal lines indicate the luminosity limit for each redshift bin and the colorbar shows the observed 𝑧-band magnitude of the UltraVISTA sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We have three luminosity complete samples for three redshift bins: For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 the luminosity limit is log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27, for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58, and for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 the luminosity limit is log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Notice that the depth of the Legacy Surveys make it possible to detect faint galaxies all the way down to log10(𝐿𝑔/𝐿 ⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 even in the upper redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In Rudnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2009), they construct luminosity func- tions for red sequence galaxies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At 𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6, they found that M★𝑔 ∼ −21 which corresponds to log10(𝐿𝑔/𝐿 ⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='43, or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 times higher than our limit here in our highest redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This means at our high redshift limit we are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 times lower than L*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3 Determination of the number of satellite galaxies The Legacy Survey 𝑔𝑟𝑧 photometry is not sufficient to achieve precise photometric redshfits for our candidate satellite galaxies and obtain- ing complete spectroscopy to these depths and over even just the EDR would be impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Therefore we quantify the satellite population around LRGs by implementing a statistical background subtraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We describe this method in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 Statistical Background Subtraction The foundation of our statistical background subtraction method con- sists of three parts: counting the number of near neighbors within a defined search radius, determining how many background sources to expect, and subtracting the second number from the first for each LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We set our search radius to correspond to the expected virial radius (𝑅200) for our LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To estimate this radius we used the stellar mass to halo mass relation and estimated the virial radius from the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The 75th and 25th percentile of LRGs masses are separated MNRAS 000, 1–14 (2022) LRG satellites 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8 redshift 6 7 8 9 10 11 log10(Lg/L ) log10(Lg/L ) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 log10(Lg/L ) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 log10(Lg/L ) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 17 18 19 20 21 22 observed zmag Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The log10(𝐿𝑔/𝐿⊙) of the UltraVISTA galaxies at z < 1 and with an observed 𝑧-band magnitude brighter than 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The solid lines denote our redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Each horizontal line marks the 90 percent luminosity complete- ness for different redshift ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The colorbar indicates the observed z-band magnitude of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This defines three luminosity complete subsam- ples: for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27, for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58, and for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 the luminosity limit is log10(𝐿𝑔/𝐿⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 dex and the median mass does not change with redshift, so we assume that LRGs have a fixed stellar mass of log10(M/M⊙) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use equations 2 and 11-14 in Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2013), which are reproduced below, where we have slightly changed the original notation for clarity: 𝑀★ 𝑀ℎ = 2𝑁 �� 𝑀ℎ 𝑀1 𝛽� + � 𝑀ℎ 𝑀1 𝛾��−1 (1) log 𝑀1(𝑧) = 𝑀10 + 𝑀11 𝑧 𝑧 + 1 (2) 𝑁(𝑧) = 𝑁10 + 𝑁11 𝑧 𝑧 + 1 (3) 𝛽(𝑧) = 𝛽10 + 𝛽11 𝑧 𝑧 + 1 (4) 𝛾(𝑧) = 𝛾10 + 𝛾11 𝑧 𝑧 + 1 (5) where 𝑀★ and 𝑀ℎ are the stellar and halo mass respectively, 𝑀1 is the characteristic mass, 𝑁 is the normalization, and 𝛽 and 𝛾 are the power law slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use best fit values for 𝑀10, 𝑀11, 𝑁10, 𝑁11, 𝛽10, 𝛽11, 𝛾10, and 𝛾11 from Table 1 in Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Using this relation we find log(𝑀ℎ/𝑀⊙) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='93, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='99, and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='04 in our three redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We therefore adopt log(𝑀ℎ/𝑀⊙) ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 for all LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Using this, we estimate 𝑅200 using the critical density (𝜌𝑐) at each epoch and the following equation: 𝑅200 = � 4𝜋(200𝜌𝑐) 3𝑀ℎ �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (6) The corresponding values of R200 for our sample is are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='57, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='59, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='64 Mpc in our three redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We therefore assume that the virial radii for our LRGs can be approximated by 𝑅200 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 Mpc and we count satellites within this radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To count the number of near neighbors N𝑛𝑛 around each LRG, we used the k-d tree algorithm from scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2011) to find sources projected within a 600 kpc radius, converted to a unique angular distance for each LRG according to Wright (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To estimate the background and foreground contamination, we create a HEALPix (Gorski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2019) map of all galaxies in the EDR using pixel dimensions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='00082 deg2 and define an annulus around each LRG between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 degree radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We determine that, at this distance, deviations from large scale structure start to dominate over the Poisson uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Because a fixed angular scale will measure different background at different redshifts, we test multiple apertures and find that our results are insensitive to background aperture size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We identify the HEALPix pixels within the background annulus and use the galaxies within those pixels as our background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This is advantageous as HEALPix allows us to very quickly index which galaxies fall in which part of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We do this separately for each LRG, which allows our background estimate to trace the local large scale structure on which each LRG lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At the median redshift of each of our redshift bins, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='45 degrees, corresponds to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='96 Mpc, well beyond the size of the largest virialized clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To minimize background and foreground contamination, we use the UltraVISTA catalog make cuts in observed (𝑔 − 𝑟) and (𝑟 − 𝑧) color for each redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We make cuts that maximize the number of galaxies in our redshift range and minimize the galaxies outside our redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For example, we find that galaxies with (𝑟 − 𝑧) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='9(𝑔 − 𝑟) and (𝑟 − 𝑧) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 are in the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We reduce background and foreground contamination by only considering galaxies within these bounds at these redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For redshift bins 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 the cuts are made at (𝑟 − 𝑧) = (𝑔 − 𝑟) and (𝑟 − 𝑧) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 and for redshift bin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 the cuts are made at (𝑟 − 𝑧) = (𝑔 − 𝑟) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 and (𝑟 − 𝑧) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' An example of these cuts can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Table 1 reports the percentage of target sources retained by the cut, as well as the percentage of sources retained that are actually outside the redshift slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This method was most effective at eliminating sources above our target redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' These cuts are implemented at all stages of the analysis, and significantly reduce the Poisson uncertainty in our background estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For each LRG, we place near neighbors, background, and fore- ground galaxies in their own color-color-magnitude diagram, using their observed (𝑟 − 𝑧) and (𝑔 − 𝑟) colors and observed 𝑧-band mag- nitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The grid consists of 50x50x50 cells, spanning -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8 < (𝑟 − 𝑧) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4, -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < (𝑔 − 𝑟) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6, and 13 < 𝑧-band magnitude < 23, corresponding to the range of our values in our catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The number of background galaxies N𝑏𝑘𝑔 in each color-color-magnitude cell is then scaled to the angular area of the 600 kpc search radius to reflect the number of background galaxies we would expect to see within that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In performing this normalization, we account for the amount of area of the sky lost due to bright stars and large foreground galaxies in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For the background, we identify Legacy Survey galaxies that are flagged for touching a Tycho-2 or Gaia star (to a 𝑔-band magnitude < 13), a large galaxy, or a star cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We determine in which HEALPix pixels in the background annulus the flagged sources are found and exclude the flagged pixel from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The median percentage of our background area with compromised photometry is less than two percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For near neighbors, we use a different method because the area of our HEALPix pixels is too large and results in the elimination of MNRAS 000, 1–14 (2022) 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Results of color-color cut for each redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The table reports the percentage of sources that are truly in the redshift bin that are retained by the cut, as well as the percentage of interloper sources retained that are actually outside the redshift slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This method was most effective at eliminating sources above our target redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Results of color-color cut redshift bin percentage of true sources retained percentage of low-z interlopers retained percentage of high-z interlopers retained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8% 11% 1 0 1 2 3 4 (g-r) color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 (r-z) color color cut boundary z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' An example of the color-color cuts made for redshift bin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 using galaxies from the UltraVISTA catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The cuts primarily eliminate galaxies that are at higher redshift than the target redshift slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' too much area within the 600 kpc search radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Furthermore, we find that the area close to our LRGs are not contaminated by bright foreground galaxies or star clusters, so the only contamination is from bright stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Rather than using HEALPix, we use the k-d tree algorithm to find Gaia DR2 stars within our 600 kpc search radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' When a star is found within the search radius, the region affected by the star’s halo is calculated by Equation 7 as prescribed by the Legacy Survey pipeline★★ 𝑅𝐺 = 150 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='511−𝑔𝑚𝑎𝑔𝐺 × (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='262/3600) (7) where R𝐺 is the resulting radius in degree and gmag𝐺 is the observed 𝑔-band magnitude for the star in the Gaia DR2 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We then use the radius to determine the area lost to contamination and remove any flagged near neighbor sources from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The median percentage of the search areas lost to contamination is less than nine percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Both of these contamination calculations are done for each LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Near neighbors and the background are then scaled by these area modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Once these area corrections are made, we calculate the number of satellites by subtracting the number of background galaxies from the number of near neighbor galaxies, N𝑠𝑎𝑡,𝑖 = N𝑛𝑛,𝑖 - N𝑏𝑘𝑔,𝑖, on a cell-by-cell basis in the color-color-magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We sum over all color-color-magnitude bins to get a total number of satellite galaxies for each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Formally, this number can be negative due ★★ https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='com/legacysurvey/legacypipe to the variation of the density of the local background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, these measurements can be used to get a statistical estimate of the number of satellite galaxies around LRGs at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To measure satellite numbers for our luminosity complete samples, we divide the LRGs into three redshift bins: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We then apply selection matrices to each resulting color- color-magnitude diagram to determine the number of satellites per LRG above the corresponding luminosity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Selection matrices are described in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 Satellite Identification in Bins of Luminosity and Redshift In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 we describe our luminosity completeness limits, derived from the UltraVISTA catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' One way to measure the rest-frame lu- minosity of our satellite galaxies would be to assume that all galaxies are at the LRG redshift and to then fit the sparsely sampled SEDs to derive rest-frame luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, the vast majority of galax- ies in close projection around any LRG are not at the LRG redshift and this would cause significant systematic errors in the luminosity (Rudnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Instead, we decide to establish a color-color dependent 𝑧-band magnitude limit, brighter than which galaxies at the LRG redshift would be above our luminosity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We map these luminosity completeness limits onto an observed 𝑧-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We perform this translation of rest-frame to observed properties, as our lack of redshift information for satellite galaxies makes it impossible to directly compute their rest-frame luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To do this, we first match the Legacy Survey catalog to UltraVISTA so we can use Legacy Survey photometry and UltraVISTA redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Then, in each (𝑟 − 𝑧) and (𝑔 − 𝑟) cell we determine the median observed 𝑧-band magnitude of UltraVISTA galaxies that are within Δ log10(𝐿𝑔/𝐿⊙) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 of the luminosity limit and in bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='05 in redshift in each color-color cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We do this for each combina- tion of luminosity limit and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This median 𝑧-band magnitude in each color-color cell defines a selection boundary in color-color space brighter than which galaxies at the redshift of the LRG are lu- minosity complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For example, if a color-color cell has a selection limit of 𝑧 = 20 at the redshift of the LRG, then only cells brighter than that limit will contribute to the total satellite count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This method assumes that all galaxies around the LRG that are in excess above the background are at the redshift of the LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The color dependence of the 𝑧-band magnitude limit accounts for the SED shape variations among satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 Distribution of Measured 𝑁sat We find the number distribution of satellite galaxies N𝑠𝑎𝑡 in each redshift and luminosity bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For LRGs at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 we find the distribution for luminosity bins log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27, log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58, and log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For LRGs in the redshift range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 we show the distribution for the log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 and log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For LRGs in MNRAS 000, 1–14 (2022) LRG satellites 7 the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 we show the distribution for the log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 luminosity bin, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' These distributions can be found in the appendix, and they represent the number distribution of possible satellite galaxies above a certain luminosity threshold for LRGs in the redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In all redshift and luminosity bins, there is a large scatter in the distribution and a tail toward higher satellite numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' An example of our N𝑠𝑎𝑡 distributions can be found in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Despite the tail of 𝑁𝑠𝑎𝑡 to high values as seen, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' in the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5 and Appendix A, the large spread in 𝑁𝑠𝑎𝑡 makes it difficult to determine at face value how many LRGs have a statistically significant detection of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To determine the significance of our N𝑠𝑎𝑡 measurement, we deter- mine how many "satellites" we would expect to detect in a random pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This constitutes the null prediction for our measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For each luminosity bin, we randomly select 10,000 galaxies from the EDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We consider these randomly selected galaxies to be our mock LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We then randomly assign each mock LRG a redshift between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We then run our statistical background subtraction method on these mock LRGs and apply the selection matrices at the mock redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' N𝑛𝑛,𝑛𝑢𝑙𝑙, N𝑏𝑘𝑔,𝑛𝑢𝑙𝑙 and N𝑠𝑎𝑡,𝑛𝑢𝑙𝑙 are determined in the same way as N𝑛𝑛, N𝑏𝑘𝑔 and N𝑠𝑎𝑡 from the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' An example of the distribution of satellites gleaned from the null test is in the second panel from the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5 and the results of the null test for all luminosity bins can be found in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A1 through A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Results of the null tests for each luminosity-complete sample are used to determine the significance of N𝑠𝑎𝑡 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We consider there to be a significant satellite detection if the measurement N𝑠𝑎𝑡 for an individual LRG exceeds the 99th percentile of satellites measured from the null result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The 99th percentile is noted on the plots of satellite numbers throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 6 we show the distribution of 𝑁𝑠𝑎𝑡 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' redshift for all of our combinations of redshift and luminosity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In each panel we denote the 99th percentile for a significant 𝑁𝑠𝑎𝑡 detection as a hori- zontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The 99th percentile is different for each combination of redshift and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At fixed luminosity (row) the threshold for a significant detection decreases with increasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' While the 99% limit decreases towards higher redshift, the fraction of LRGs with a significant number of satellites is constant within the uncer- tainties, as shown in the third row of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This likely stems from the different Poisson uncertainty from the background subtraction in different redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' On one hand, our constant luminosity thresh- old will correspond to a brighter 𝑧-band magnitude limit at lower redshift, which should decrease the number of interlopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' On the other hand, our color cuts described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 are very effective at removing high redshift (and likely fainter) interlopers, but reject the lowest fraction of interlopers for our lowest redshift bins and would result in a higher contribution to the uncertainty in 𝑁𝑠𝑎𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The combi- nation of these effects is likely responsible for the modest dependence of the 99% confidence level on redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Most LRGs in all bins do not have a significant satellite mea- surement within a 600 kpc radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Some LRGs have small negative number of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This is a result of the statistical nature of our sub- traction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The existence of negative satellites - an obviously unphysical phenomenon - is a reflection of the random fluctuations in the background measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At times, those fluctuations will produce a high background when compared to near neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 Color Distribution of LRG Satellites While our method makes it impossible to study individual satellite galaxies in detail, it is possible to determine color and magnitude 40 20 0 20 40 60 80 100 0 5 10 counts mean = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 median = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='014 percentage above 99% level: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% Data 40 20 0 20 40 60 80 100 0 20 40 60 80 counts mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8689 median = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4079 KS pval = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='217e-09 Null 40 20 0 20 40 60 80 100 0 20 40 60 80 counts mean = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='254 median = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='297 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0008567 percentage above 99% level: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='9% Nsat, mod = 1 model 40 20 0 20 40 60 80 100 0 20 40 60 80 counts mean = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='205 median = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='398 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2029 percentage above 99% level: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% Nsat, mod = 3 model 40 20 0 20 40 60 80 100 Number of Satellites 0 20 40 60 80 counts mean = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='937 median = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='218 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='002221 percentage above 99% level: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% Nsat, mod = 7 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data (top panel), the null test (second panel), and models for different intrinsic ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑 ⟩ numbers for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 and log10(𝐿𝑔/𝐿⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 (bottom three panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates the 99th percent confidence for each bin, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Above this line, the number of satellites would only occur less than 1 percent of the time in a randomly drawn sample of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The null test and the models are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 and 7 illustrate models that are not consistent with the data, with too few and with a median number of satellites that is too low and too high, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑 ⟩ = 3 shows a model that is consistent with the data, as illustrated with a p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' All panels show a skewed distribution, with a tail towards high satellite counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, the data and models show a more pronounced tail than the null prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The plots also indicate the percentage of LRGs in both the data and the models that exceed the number of satellites at 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' distributions as a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 7 illustrates the observed (𝑟 − 𝑧) color vs observed 𝑧-band magnitude of satellites compared to LRGs in bins of redshift and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We show the distribution of LRGs and satellites for the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In general, the satellite population is fainter than the population of LRGs, though there is some overlap in the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In all cases satellite galaxies appear to have (𝑟 − 𝑧) colors similar to or bluer than that of LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To quantify the difference in observed (𝑟 − 𝑧) color between LRGs and their satellites, we implement the following process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We cannot identify an individual galaxy as a satellite, hence we do not know the exact satellite (𝑟 − 𝑧) colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' As an approximation, we use the midpoint value of the (𝑟 − 𝑧) color bins as satellite colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Each (𝑟 − 𝑧) color bin is weighted by the number of galaxies in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We then subtract this color from the color of the host LRG and we interpolate the resulting Δ(𝑟 − 𝑧) onto a reference grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We show the distribution of Δ(𝑟 − 𝑧) in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 8 for each redshift and luminosity bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The right panel shows the medians of the distributions for each luminosity and redshift bin with the error MNRAS 000, 1–14 (2022) 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 40 20 0 20 40 60 80 100 Nsat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 99 percent confidence = 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 40 20 0 20 40 60 80 100 Nsat log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 99 percent confidence = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 99 percent confidence = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 redshift 40 20 0 20 40 60 80 100 Nsat log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 99 percent confidence = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='50 redshift 99 percent confidence = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 99 percent confidence = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='46 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The number of satellites for each LRG plotted against the LRG redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The horizontal lines indicate the 99th percent confidence for each bin, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Above this line, the number of satellites would only occur less than 1 percent of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fewer satellites are needed for a significant detection in the high redshift bins compared to the lower redshift bins, at fixed luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Table 2 gives the proportion of LRGs in each bin that have significant satellite detections, as well as the upper and lower bounds given by a 99% binomial confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Despite the lower threshold, at fixed luminosity the proportion of LRGs with significant detections decreases with increasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' bars representing the 68 percent confidence interval in the weighted median determined by bootstrap resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 8 shows no statistical evidence for redshift evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 Forward Modeling the Satellite Population With the above techniques we can infer the number of LRGs with a statistically significant excess of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, because of the large spread in 𝑁𝑠𝑎𝑡 and other potential systematics (see previous sections), we cannot straightforwardly determine the average number of satellites per LRG using only the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For this reason we develop a method in which we use our null distribution of 𝑁𝑠𝑎𝑡 to forward model the predicted distribution of the measured 𝑁𝑠𝑎𝑡 for an intrinsic distribution of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Modeling in this way attempts to answer the question, if we integrate down the satellite galaxy luminosity function to a certain luminosity and at a certain redshift, would we find that the number distribution of satellite galaxies around LRGs to be consistent with LRGs having an intrinsic number of satellites?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We start with the assumption that the mock LRGs in a given model have an intrinsic mean number of satellites, with the number around each mock LRG independently drawn from a Poisson distribution with the corresponding expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This number is effectively the integral of the luminosity function of satellites down to the lumi- nosity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For example, a model with an expectation value of ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 would have all mock LRGs populated with a number of satellites drawn from a Poisson distribution with that expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) LRG satellites 9 0 1 2 (r-z) color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 LRG 0 1 2 (r-z) color log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 LRG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 LRG 16 17 18 19 20 21 22 observed z-band magnitude 0 1 2 (r-z) color log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 LRG 16 17 18 19 20 21 22 observed z-band magnitude LRG 16 17 18 19 20 21 22 observed z-band magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 LRG 0 20 40 60 80 100 120 140 counts 0 20 40 60 80 100 120 140 counts 0 20 40 60 80 100 120 140 160 counts 0 20 40 60 80 100 120 140 counts 0 25 50 75 100 125 150 175 counts 0 20 40 60 80 100 120 140 160 counts Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Observed (𝑟 − 𝑧) color vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 𝑧-band magnitude diagram for LRGs (red dots) and satellite galaxies (shading) for different redshift and luminosity bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Each red dot represents an individual LRG in this redshift range and the shaded regions represent the distribution of satellite galaxies in bins of (𝑟 − 𝑧) observed color and observed 𝑧-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The vertical lines across the top of the first panel represents the median uncertainty in the color measurement at the observed 𝑧-band magnitude at which they are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In general, the satellite population is fainter than the population of LRGs, though there is some overlap in the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In all cases satellite galaxies appear to have (𝑟 − 𝑧) colors similar to or bluer than that of LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We quantify this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 (r-z) 0 250 500 750 1000 1250 1500 1750 counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='55 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='17 median of (r-z) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Left: The distribution of the difference between the observed (r-z) colors of satellites and their host LRGs, (𝑟 − 𝑧)𝐿𝑅𝐺 - (𝑟 − 𝑧)𝑠𝑎𝑡 = Δ(𝑟 − 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Colors represent different luminosity limits and line styles represent different redshift ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Right: The median value of Δ(𝑟 − 𝑧) for each luminosity and redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Error bars represent 16th to 84th percentile in the weighted median determined through boostrap resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Nsat, mod log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 log10(Lg/L ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 Tal+2012 SAMs 0 2 4 6 8 10 Nsat, mod 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 redshift 0 2 4 6 8 10 Nsat, mod Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A summary of our forward modelling results, described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The y-axis is the number of model satellites and the x-axis is redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The shaded regions show the range of numbers of satellites with a p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='05 and the dashed boxes show the range with a p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Each panel includes our forward modeling results as rectangles and results from an analysis of the satellite counts from semi-analytic models of Hirschmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2016) as square points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The error bars for the SAMs are the 68% confidence limits on the distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the errors in the mean are smaller than the points and are thus not plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The bottom panel also includes the satellite numbers for two redshift bins from Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2012), shown as round black points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' These are scaled to our search radius of 600 kpc and the error bars represent the 68% confidence limits determined by the range of 𝛼𝑠 in Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) LRG satellites 11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Table summary of satellite count and forward modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' N𝐿𝑅𝐺, N𝐿𝑅𝐺(sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' satellites), and N𝐿𝑅𝐺(sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' satellites)/N𝐿𝑅𝐺 are the total number of LRGs, the total number of LRGs with a significant satellite detection, and the proportion of LRGs with significant satellite counts with binomial confidence intervals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The row labeled “⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ range” are the range of values of ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ drawn from a Poisson distribution that are consistent with the data to the 95% confidence (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='75% confidence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Summary Table of Satellite Counts and Forward Modeling log10(L𝑔/L⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 log10(L𝑔/L⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 log10(L𝑔/L⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 ≤ z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 N𝐿𝑅𝐺 309 309 617 309 617 897 N𝐿𝑅𝐺(sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' satellites) 14 16 16 14 15 12 N𝐿𝑅𝐺(sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' satellites)/N𝐿𝑅𝐺 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='006 ⟨ N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ range 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5) After we determine how many satellites to add to each mock LRG, we add sources to the near neighbor matrix N𝑛𝑛,𝑛𝑢𝑙𝑙 in cells that are at least as bright in the median observed 𝑧-band magnitude for the color-color cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In practice, these galaxies are randomly chosen duplicates of galaxies already in those cells, to ensure that the added mock satellites follow the same color-color-magnitude distribution of the actual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The resulting matrix is N𝑛𝑛,𝑚𝑜𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The background matrix for the mock LRG remains unmodified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model number of satellites, N𝑠𝑎𝑡,𝑚𝑜𝑑, is calculated by subtracting N𝑏𝑘𝑔,𝑛𝑢𝑙𝑙 from N𝑛𝑛,𝑚𝑜𝑑, and then applying the appropriate selection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A sample of our model results is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5, which indicates the mean, median, and p-value from the Komogorov-Smirnov 2- sample test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The p-value indicates whether or not the model and data are consistent with being drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Small p-values indicate that the model and the data likely did not come from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For example, the case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5 for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 and log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 has an acceptable p-value (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='003) for the model with ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3, but significantly lower p-values for all other ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩, including the null result of zero mean satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The remaining comparisons between the data, null result, and models can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 9 shows a summary of our model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The solid boxes represent the range of satellite numbers expected for p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='05 and the dashed boxes represent the range for p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The colors represent the same luminosity bins as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Based on these results, we can rule out a lack of evolution to 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 at a fixed luminosity limit at the greater than 95% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We conclude this as the allowed range of ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ at 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 in each luminosity bin do not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We cannot reject the no-evolution hypothesis at the greater than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7% level as the allowed values at that confidence level overlap across all redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We do not have the precision to constrain the faint end of the luminosity function, but the lack of a clear increase in ⟨N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ with decreasing luminosity limit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 9 implies a flat or declining faint-end luminosity slope for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A summary of these result can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5 Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 Colors and Numbers of LRG satellites While our statistical subtraction methods leaves us unable to study any single satellite galaxy in particular, nevertheless we are able to study satellite properties on a population level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 8 illustrates the difference in the color distribution between satellites and their host LRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The color difference across all bins are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The color distribution for the highest redshift and highest luminosity bin has a significantly larger spread than the other bins and extends to fainter luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The origin of this is seen in the bottom right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 7 in which we see that the LRGs and the reddest satellites are significantly redder than in lower redshift bins, though the satellites still reach to as blue colors as in the other redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This behavior stems from the color- dependent k-corrections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the observed (𝑟 − 𝑧) color of passive galaxies becomes rapidly redder with increasing redshift while the observed color for blue galaxies changes much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The result is an increase in the difference between the reddest and bluest objects in observed color as one looks to higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 9, at a fixed redshift we see substantial overlap between acceptable models for each luminosity limit, such that we cannot rule that the mean number of satellites is independent of the luminosity limit, at least over our luminosity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Despite that, there are some hints of a dependence on the luminosity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In the low redshift bin we can see that the 2− and 3 − 𝜎 lower bounds of acceptable models increases as the luminosity limit decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27, the lower bound is ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The lower 3 − 𝜎 bound for log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 is ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 and for log10(𝐿𝑔/𝐿 ⊙) ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This is somewhat as expected, as we should be increasing the number of satellites as we integrate further down the luminosity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3, there is tentative evidence for evolution at 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 in satellites selected to a fixed luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Interpreting this evolution in terms of the true number of satellites down to a given mass limit is complicat ed, as satellites will evolve in 𝑀/𝐿★ over time and as such should on average be reducing their luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This implies that satellites which are just visible above our luminosity limit at higher redshift may fade below our limit by a lower redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The tentative evolution we see may therefore be indicative of an even stronger evolution in the number of satellites to a fixed stellar mass threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We will address this issue in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We compare our results to Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2012, hereafter T12) by integrating the luminosity function they found for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The T12 analysis differs from ours in important ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' First, they only go down to a luminosity of log10(𝐿𝑔/𝐿 ⊙) ∼ 10 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='28 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4 and log10(𝐿𝑔/𝐿 ⊙) ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7, whereas this work goes down to log10(𝐿𝑔/𝐿 ⊙ ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 out to 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Second, T12 measures satellites out to 1 Mpc, which is a larger radius than we search in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To account for these differences we first integrate the luminosity function from T12 down to their luminosity limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' These limits are brighter than ours but from a small sample at lower redshift T12 showed that the luminosity function starts flattening below log10(𝐿𝑔/𝐿 ⊙ ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We therefore do not think that the T12 satellite numbers when integrated to our luminosity limits would significantly increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We then account for the different in radial search areas by scaling the T12 result by the ratio of the mean N𝑠𝑎𝑡 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6 Mpc and 1 Mpc from our own data MNRAS 000, 1–14 (2022) 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The T12 points are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The error bars represent the 68% confidence limits determined by the range of 𝛼𝑠 in T12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The T12 points are in broad agreement with our data though they appear to have a slightly higher number of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Additionally, we also compare our results to semi-analytic models based on (Hirschmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 2016, private communication)††.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We measure the number of satellites around simulated LRGs out to a 3D radius of 600 kpc in bins of redshift and 𝑔-band luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We plot these model predictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There is good agreement between our results and the semi-analytic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At a fixed redshift, the SAM has the same trend with luminosity limit as in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' How- ever, they show a shallower trend with redshift at a fixed luminosity limit than does our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' It is not clear if this difference lies in the approximations we make in our comparison, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the use of 3D radii for the models, or if it reflects a true problem with the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We will treat this in more depth in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 The Future Mass Growth of LRGs We next estimate the amount of mass that can be gained by each LRG resulting from satellite accretion at 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We cannot measure the mass of individual satellite galaxies, but can use their statistical color and luminosity distribution relative to that of the host LRGs to compute the average amount of mass in satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We start with the Δ(𝑟 − 𝑧) and Δzmag, the average difference between the observed 𝑧-band magnitude for LRGs and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use EZgal (Mancone & Gonzalez 2012) to output a model from Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2009) with a Chabrier initial mass function (Chabrier 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We used an exponentially declining SFH from this model to define a relationship between observed (𝑟 − 𝑧) color and the ratio of the stellar mass (𝑀★) to the observed 𝑧-band flux ( 𝑓𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This was done at the mean observed redshift 𝑧𝑜𝑏𝑠 of each of our LRG subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In this model the relation between color and log10(𝑀★/ 𝑓𝑧) represents a trend with age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This relation is valid for all smooth SFHs and dust extinction moves galaxies relatively parallel to this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Using the mean difference in observed color between LRGs and their satellites Δ(𝑟 − 𝑧), we use this relation to infer a difference in log10(𝑀★/ 𝑓𝑧) between LRGs and satellites, which we call Δ log10(𝑀★/ 𝑓𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' From this difference and the ratio of the mean 𝑧-band flux between LRGs and satellites we can compute the ratio of the stellar mass in the two populations as 𝑙𝑜𝑔10 � 𝑀★,𝐿𝑅𝐺 𝑀★,𝑠𝑎𝑡 � = Δ𝑙𝑜𝑔10 � 𝑀★ 𝑓𝑧 � − 𝑙𝑜𝑔10 � 𝑓𝑧,𝑠𝑎𝑡 𝑓𝑧, 𝐿𝑅𝐺 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (8) We make a number of assumptions and simplifications in this cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' First, we assume all satellites have the same star formation history or can be represented by a single star formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In reality, the satellites with lower luminosity may have different star formation histories but at the same time all smooth SFHs should lie in the same plane of log10(𝑀★/ 𝑓𝑧) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (𝑟 − 𝑧)𝑜𝑏𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Additionally, we assume that the star formation histories of LRGs and satellites are the same and we assume that the effects of dust are negligible on the relation we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We find that log10 � 𝑀★,𝐿𝑅𝐺 𝑀★,𝑠𝑎𝑡 � ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='77 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='22, with the highest mass in satellites occurring in our highest redshift and highest luminosity limit bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The lowest mass in satellites occurs in our lowest redshift and lowest luminosity limit bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At a given redshift, log10 � 𝑀★,𝐿𝑅𝐺 𝑀★,𝑠𝑎𝑡 � is constant to within ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='1 dex for satellites with different luminosity limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Furthermore, at the same luminosity limit, †† https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='com/inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='it/gaea/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='pli=1 but over different redshifts, log10 � 𝑀★,𝐿𝑅𝐺 𝑀★,𝑠𝑎𝑡 � increases as redshift increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The increase in relative satellite mass is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 dex between the two lowest redshift bins for the log10(𝐿𝑔) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 limit and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3 dex between the two redshift lowest bins for the log10(𝐿𝑔) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There is no increase in relative satellite mass between the middle and highest redshift bin for the log10(𝐿𝑔) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There are two reasons why the relative mass in satellites might increase to higher redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' First, it could be that there are more satel- lites at higher redshift and that the rate of consumption by the LRGs overtakes the accumulation of new satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Second, it could be that all satellites will move to lower mass-to-light ratio values toward higher redshift because they are younger, so a fixed luminosity limit will probe to a lower satellite stellar mass limit at higher redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We cannot distinguish between these two possibilities without measur- ing the rest-frame colors of satellites or their stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' However, given these caveats we can say that LRGs can at most grow by ∼ 15% from 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='575 to the present day and by ∼ 6% from 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='275 to the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This is consistent with the findings from Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' (2008), which determined that the mass of massive red galaxies grows by 30% since z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 6 Summary and Conclusions In this study, we use the deep photometry from the DESI Legacy Imaging Surveys to characterize the satellite population around SDSS-identified luminous red galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' SDSS LRGs represent a ho- mogeneous population of massive red ellipticals, ideal for investigat- ing the mechanisms by which the mass of massive ellipticals builds up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We use the Legacy Survey photometry for SDSS DR14 LRGs froom the LOWZ and CMASS samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The deeper Legacy Survey photometry allows us to carry out a characterization of LRG satellites to fainter luminosities and higher redshifts than previously possible for a large sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We have 1,823 LRGs in our 25 square degree sample area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We perform a statistical background subtraction in (𝑔 − 𝑟), (𝑟 − 𝑧), and 𝑧-band magnitude space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To determine what counts as a signif- icant satellite detection, an LRG must have a number of satellites above the 99th percentile of the null result in each luminosity bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We develop a technique to forward model the satellite distribution, accounting for the survey selection function and systematics in our background subtraction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This method deals with the lack of a priori knowledge of the redshifts and luminosities of the satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The main results of this paper are as follows: LRGs are in general brighter than their satellites in the observed 𝑧-band, but satellite galaxies have similar or slightly bluer observed (𝑟 − 𝑧) colors to LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This is illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 6 and in Table 2 that the proportion of LRGs that have significant satellite detections decreases with increasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This does not have a straightforward interpretation with regards to the true population of satellites, as the threshold for sig- nificance varies with redshift and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' To better interpret the number of satellites around the mean LRG in each bin we developed a forward modeling approach in which we use mock samples to simulate the observed signature of difference satellite populations, assuming that the total number of satellites is drawn from a Poisson distribution with expectation value ⟨ N𝑠𝑎𝑡,𝑚𝑜𝑑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' For LRGs at all combinations of redshift and rest-frame 𝑔-band satellite luminosity limit, we can strongly rule out a lack of satellites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' At fixed redshift, the 3-𝜎 lower bounds go toward lower accept- able 𝑁𝑠𝑎𝑡,𝑚𝑜𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' There is, however, significant overlap in acceptable MNRAS 000, 1–14 (2022) LRG satellites 13 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This implies a flat or declining faint-end luminosity slope for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In our two highest luminosity bins we find tentative evidence for evolution in ⟨ N𝑠𝑎𝑡,𝑚𝑜𝑑⟩ out to 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 though cannot rule out the no-evolution hypothesis as the greater than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The interpretation of this evolution is complicated by the expected 𝑀/𝐿★ evolution of satellite galaxies over this redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We calculated that LRGs can at most grow by ∼ 15% from 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='575 to the present day and by ∼ 6% from 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='275 to the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This pilot program has demonstrated the ability of the DESI Legacy Imaging Survey, coupled with LRG spectroscopy, to probe the satellite properties of the most massive galaxies in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Despite the larger area of the EDR, it is not practical in our sample to investigate the trends of satellite galaxies as a function of LRG properties such as stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' In a future work we will apply these techniques to the full DESI Legacy Imaging Survey, in order to ana- lyze the full population of SDSS-identified LRGs and to investigate the luminosity function of satellite galaxies and their dependence on LRG property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Acknowledgements The authors would like to thank John Moustakas at Siena College for calculating the masses of LRGs used in this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' We thank Arjun Dey and Stephanie Juneau for useful discussions that helped to spawn this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The authors acknowledge the support of NSF AST- 1716690, AST-1517815, NASA ADAP-80NSSC19K0592, ADAP- NNX17AF25G, NASA HST-AR-14310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='001-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' They also acknowl- edge the support of the International Space Sciences Institute in Bern, which hosted discussions of this work as part of the “The Effect of Dense Environments on Gas in Galaxies over 10 Billion Years of Cosmic Time” and “COSWEB: The Cosmic Web and Galaxy Evolu- tion” teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' acknowledges the support of a University Graduate Fellowship and the Lowry Graduate Fellowship from the University of Kansas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' thanks the Lorentz Center in Leiden for support via their workshop “Galaxy Evolution in the Cosmic Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='R, acknowledges the support of an ESO visiting science fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The Legacy Surveys consist of three individual and complemen- tary projects: the Dark Energy Camera Legacy Survey (DECaLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Proposal ID #2014B-0404;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' NOAO Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' ID #2015A-0801;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Sur- vey (MzLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' ID #2016A-0453;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' PI: Arjun Dey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIR- Lab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Bok telescope, Steward Observatory, University of Arizona;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' and the Mayall telescope, Kitt Peak National Observatory, NOIRLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' NOIRLab is operated by the Association of Universities for Re- search in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Funding for the DES Projects has been provided by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Department of Energy, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' National Science Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Ministry of Science and Education of Spain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Science and Technology Facilities Council of the United Kingdom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Higher Education Funding Council for England,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the National Center for Supercomputing Applications at the University of Illinois at Urbana- Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Kavli Institute of Cosmological Physics at the Uni- versity of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Center for Cosmology and Astro-Particle Physics at the Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Financiadora de Estudos e Projetos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fundacao Carlos Chagas Filho de Amparo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fi- nanciadora de Estudos e Projetos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Conselho Na- cional de Desenvolvimento Cientifico e Tecnologico and the Minis- terio da Ciencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Tecnologia e Inovacao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Deutsche Forschungs- gemeinschaft and the Collaborating Institutions in the Dark Energy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The Collaborating Institutions are Argonne National Labo- ratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of California at Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Centro de Investigaciones Energeticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Medioambien- tales y Tecnologicas-Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the DES-Brazil Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Ed- inburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Eidgenossische Technische Hochschule (ETH) Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Fermi National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Institut de Ciencies de l’Espai (IEEC/CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Institut de Fisica d’Altes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' NSF’s NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Ohio State Uni- versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' the Uni- versity of Sussex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' and Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Prior- ity Research Program “The Emergence of Cosmological Structures” Grant #XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The BASS is also supported by the Exter- nal Cooperation Program of Chinese Academy of Sciences (Grant #114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #11433005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Insti- tute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' NEOWISE is funded by the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' DE-AC02- 05CH1123, by the National Energy Research Scientific Comput- ing Center, a DOE Office of Science User Facility under the same contract;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' and by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' National Science Foundation, Division of Astronomical Sciences under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' AST-0950945 to NOAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Some of the results in this paper have been derived using the healpy and HEALPix packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Data Availability The data analyzed in this paper come from DR8 of the DESI Legacy Imaging Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' These data are available to the public at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 709, 1018 van de Sande J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 2013, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 771 van der Wel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Holden B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Zirm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Franx M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Rettura A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Illingworth G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', Ford H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 2008, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 688, 48 van der Wel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 2014, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=', 788 A Satellite Distribution Modeling Here we show the results of our LRG satellite models based on a Poisson distribution in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A1 through A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Models that fit the data with a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='05 are shown, as well as examples of models that were not good fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) LRG satellites 15 50 0 50 100 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 counts mean = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='894 median = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='12 percentage above 99% level: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5% Data 50 0 50 100 150 0 20 40 60 80 counts mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='25 median = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='182 KS pval = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='121e-08 Null 50 0 50 100 150 0 20 40 60 80 counts mean = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='221 median = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='782 KS pval = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='878e-05 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2% Nsat, mod = 2 model 50 0 50 100 150 Number of Satellites 0 20 40 60 80 counts mean = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='768 median = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='997 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5436 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7% Nsat, mod = 8 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data, the null test, and examples of Poisson models for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 and log10(L) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A description of how models are formulated can be found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The distribution the data is compared to the null test and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 8 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 2 is included as an example of model that does not have a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 50 25 0 25 50 75 100 125 0 5 10 15 counts mean = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='656 median = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='842 percentage above 99% level: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2% Data 50 25 0 25 50 75 100 125 0 20 40 60 80 counts mean = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='051 median = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='576 KS pval = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='054e-07 Null 50 25 0 25 50 75 100 125 0 20 40 60 80 counts mean = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='099 median = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='546 KS pval = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='234e-05 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2% Nsat, mod = 1 model 50 25 0 25 50 75 100 125 Number of Satellites 0 20 40 60 80 counts mean = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='593 median = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='095 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='436 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='8% Nsat, mod = 6 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 and log10(L) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A description of how models are formulated can be found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates the 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The distribution the data is compared to the null test and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 6 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 is included as an example of model that does not have a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) LRG satellites 17 20 0 20 40 60 80 100 0 5 10 15 20 25 counts mean = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='54 median = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='754 percentage above 99% level: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6% Data 20 0 20 40 60 80 100 0 50 100 150 counts mean = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='301 median = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='07083 KS pval = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='362e-07 Null 20 0 20 40 60 80 100 0 50 100 150 counts mean = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='197 median = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='051 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4182 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% Nsat, mod = 3 model 20 0 20 40 60 80 100 Number of Satellites 0 50 100 150 counts mean = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='043 median = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='864 KS pval = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='045e-05 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='6% Nsat, mod = 5 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 and log10(L) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A description of how models are formulated can be found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates the 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The distribution the data is compared to the null test and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 5 is included as an example of model that does not have a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 50 25 0 25 50 75 100 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 counts mean = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='311 median = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='639 percentage above 99% level: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5% Data 50 25 0 25 50 75 100 125 0 20 40 60 80 counts mean = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='798 median = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='296 KS pval = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='855e-07 Null 50 25 0 25 50 75 100 125 0 20 40 60 80 counts mean = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='806 median = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='197 KS pval = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='633e-05 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2% Nsat, mod = 1 model 50 25 0 25 50 75 100 125 Number of Satellites 0 20 40 60 80 counts mean = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='852 median = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='577 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7454 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7% Nsat, mod = 7 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 and log10(L) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A description of how models are formulated can be found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates the 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The distribution the data is compared to the null test and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 7 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 1 is included as an example of model that does not have a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) LRG satellites 19 20 0 20 40 60 80 0 5 10 15 20 25 counts mean = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='615 median = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='601 percentage above 99% level: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% Data 20 0 20 40 60 80 0 50 100 150 counts mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='2848 median = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7713 KS pval = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='761e-06 Null 20 0 20 40 60 80 0 50 100 150 counts mean = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='115 median = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='951 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7539 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4% Nsat, mod = 3 model 20 0 20 40 60 80 Number of Satellites 0 50 100 150 counts mean = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='068 median = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='003 KS pval = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='047e-05 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='9% Nsat, mod = 5 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='35 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 and log10(L) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A description of how models are formulated can be found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates the 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The distribution the data is compared to the null test and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 5 is included as an example of model that does not have a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' 20 0 20 40 60 0 10 20 30 counts mean = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='183 median = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='615 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3% Data 20 0 20 40 60 0 50 100 150 200 counts mean = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='671 median = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='7421 KS pval = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='687e-18 Null 20 0 20 40 60 0 50 100 150 200 counts mean = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='084 median = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='155 KS pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='4381 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='3% Nsat, mod = 2 model 20 0 20 40 60 Number of Satellites 0 50 100 150 200 counts mean = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='073 median = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='183 KS pval = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='111e-05 percentage above 99% level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5% Nsat, mod = 3 model Nsat, mod Histograms for log10(Lg/L ) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 Figure A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' Distributions of the number of satellites found from the data, the null test, and Poisson models for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='65 and log10(L) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' A description of how models are formulated can be found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The dot-dash line indicates the 99 percent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The distribution the data is compared to the null test and the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' The model ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 2 is included as an example of a model that is a good fit to the data, according to the 2-sample K-S test, and ⟨𝑁𝑠𝑎𝑡,𝑚𝑜𝑑⟩ = 3 is included as an example of model that does not have a good fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E4T4oBgHgl3EQfrw0X/content/2301.05210v1.pdf'} diff --git a/hdE1T4oBgHgl3EQfMgPQ/vector_store/index.faiss 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Thomas1 and Harsha S. Bhat2 +i. Institut des Sciences de la Terre Paris, Sorbonne Universit´e, CNRS-UMR 7193, Paris, France. +ii. Laboratoire de G´eologie, ´Ecole Normale Sup´erieure, CNRS-UMR 8538, PSL Research Uni- +versity, Paris, France. +1 +Friction Laws +1.1 +Historical notions about friction +Friction is resistance to motion that appears when two surfaces in contact slide against one +another. Generally speaking, the concept of ‘friction’ describes the dissipation of energy that +occurs. Most phenomena associated with sliding friction can be understood from observations +made by Leonardo da Vinci. He was the first to note that, based on his experiments, friction +is proportional to 1/4th of the applied pressure and that it is independent of the area of +contact between two active surfaces. This latter observation was inspired by the fact that +the resistance to sliding of a coil of rope is the same as for a stretched piece of rope. +Almost two centuries later, in the 18th century, Guillaume Amontons and Charles- +Augustin de Coulomb, carried out rigorous experiments on friction, with the aim of ob- +taining quantitative results. The collective work by L. da Vinci, G. Amontons and C.-A. de +Coulomb led to the two fundamental ’laws’ of friction. These statements, simple and still +valid, are widely applicable: +• +the friction force acting between two sliding surfaces is proportional to the load pressing +the surfaces together. That is, these two forces have a constant ratio, often called the +coefficient of friction. +• +the sliding force is independent of the apparent area of contact between the two surfaces. +The discoveries that followed (cf. Chapter 1), led researchers to revisit these laboratory +experiments in order to better understand earthquakes. In 1966, in a now-famous paper, +Brace and Byerlee showed that the creation of new fractures was not the only model that +could explain the existence of seismic faults Brace & Byerlee (1966). In their experimental +protocol, they pre-cut a rock sample and loaded its extremities, while also applying confining +pressure. They observed that the sliding between the two pieces of rock was not continuous, +but a jerky motion with accelerations and decelerations. This was the origin of the theory, +which is widely accepted today, that earthquakes are governed by frictional forces. +1.2 +From static friction to dynamic friction +If we go back to the fundamental laws of friction stated by Amonton-Coulomb, they are +mathematically expressed as follows. The frictional force Ffric = τA is independent of the +contact area A (τ being the shear stress). Ffric is proportional to the applied normal force +arXiv:2301.01605v1 [physics.geo-ph] 3 Jan 2023 + +2 +Fn = σeffA through the constant µ (σeff corresponds to the effective normal stress). We +thus have: +µ = Ffric +Fn += +τ +σeff +(1) +Let us now consider an object of mass M placed on a table. The force Fn = Mg is, therefore, +normal to the surface. We apply a tangential force Ft parallel to the surface of the table. If +the object is initially at rest, a motion may be produced if a force Ft, greater than Ffric, is +applied. In this case, the coefficient µs is called the coefficient of static friction. +Ffric = Fs = µsFn +(2) +Now, if the object is displaced at a finite velocity over the surface, it has been experimentally +found that the frictional force is also proportional to the normal force, through the coefficient +µd, called the coefficient of dynamic friction: +Ffric = Fd = µdFn +(3) +Early experiments showed that the coefficient of static friction is different from the coeffi- +cient of dynamic friction Rabinowicz (1958). Static friction has the property of increasing +logarithmically with time, and dynamic friction depends on the velocity V . +From the classic work carried out by Kostrov Kostrov (1964, 1966) and Eshelby Eshelby +(1969), it soon became clear that friction also played a fundamental role in the initiation, +rupture development and ‘healing’ of faults. The classic Amonton-Coulomb model, however, +led to an impasse. Among other physical problems, it postulated the hypothesis of an instan- +taneous modification of the coefficient of friction, from its static value to its dynamic value. +This brings in singularities (infinite stresses) at the rupture front (red model in figure 1). +This model lacks a scale of length that makes it possible to define a finite quantity +of energy released at the rupture front. There are two possible options. One consists of +defining the characteristic quantity of slip (between the two surfaces) required to move +from static friction to dynamic friction. The other consists of introducing a characteristic +time in which friction decreases from µs to µd. In this second case, a scale of characteristic +length emerges when the characteristic time is related to the slip velocity. For example: to +explain his experiments on friction, Rabinowicz Rabinowicz (1958) introduced the concept +of a ”critical distance” dc during which the gap between the static friction and the dynamic +friction is closed. He related this critical distance to the velocity, V = Dc/tw. Here tw is +called weakening time. +In general, the laws called weakening friction laws were thus developed to reproduce +seismic behavior. We speak of weakening because the friction reduces with the slip (or rate +of slip) and these laws can thereby produce instabilities Bocquet (2013); Zhuravlev (2013); +Romanet (2017). This ingredient is required to anticipate seismic velocities (m/s) in the +models. We will now present the most used models in the following sections. +1.3 +Slip weakening friction law +In fracture mechanics, the model where friction weakens with distance, also known as the +cohesive zone model, postulates that: + +3 +position +Stress +c) +position +Slip +b) + +x2 +x1 +μd +slip +a) +Friction +A +B +δc +μs +? +position +cohesive zone +crack +x2 +x1 +∞ +Figure 1. Comparison between the rupture model hypothesizing linear elasticity (red curve) and +the cohesive zone model (dotted blue curve). a) Coefficient of friction in terms of the quantity of +slip. b) Quantity of slip in terms of the position along the fracture. The point x1 is in the position +A on the friction curve and the point x2 is at position B. c) Stress field close to the rupture front. +• +the rupture process, which causes the shift from static friction to dynamic friction, is +confined to the fracture plane, +• +inelastic deformation begins when the stresses on the rupture front reach a certain +critical level, +• +we reach the value of the coefficient of dynamic friction when the displacement on the +fracture plane exceeds a critical value δc Leonov & Panasyuk (1959); Barenblatt (1959); +Dugdale (1960). +This law was introduced in the context of a study of tension fractures, in order to solve the +problem of singularities coming up (infinite stresses) on the rupture front (blue model in +figure 1). +The slip weakening friction law was introduced by Ida Ida (1972) and Andrews Andrews +(1976) to model dynamic ruptures for 2D models, and by Day Day (1982) for 3D models. +This is analogous to the cohesive zone model, but for mode II fractures, that is, for shear +fractures. In this law, the slip is zero until the shear stress τ reaches a maximum value +(elasticity limit) that will be denoted by τ s +f. Once this stress is attained, the slip starts and +the resistance to the sliding τf decreases linearly until the value τ d +f , i.e., when the plane has + +4 +Slip +Friction +μs +μd +δc +Slip Velocity +Friction +μs +μd +a) +b) +Figure 2. Schematic illustration of (a) the slip weakening friction law, (b) the velocity weakening +friction law +slipped with a critical value δc: +τf(δ) = +� +� +� +(τ s +f − τ d +f ) +� +1 − δ +δc +� ++ τ d +f +; δ < δc +τ d +f +; δ > δc +(4) +If this law is combined with the Amonton-Coulomb law (equation 1), we have: +τf(δ) = +� +� +� +� +(µs − µd) +� +1 − δ +δc +� ++ µd +� +σeff +; δ < δc +µdσeff +; δ > δc +(5) +where µd < µs. In their article, Palmer and Rice Palmer & Rice (1973) presented a law +that is very close to this for which they could derive a complete analytical solution for the +rupture front. They showed that this law made it possible to regularize the numerical model +by distributing the stresses and the slip over a distance controlled by the length scale in the +friction law. +A few nuanced but important points with respect to the slip weakening law: +i. This friction law describes the start and growth of a seismic rupture. The more the fault +slips, the weaker its resistance. If the shear stress on the fault, τ, is uniform, then this +law implies that the fault will continue to slip indefinitely until τ < τf. This does not +match the observations. There are therefore two possibilities: either τ is heterogeneous +along the the fault due to its geometric complexity (branches, non-linear plane, fault +jump etc.) or related to past earthquakes. The second possibility, since faults have finite +length, is that the rupture stoped because the earthquake ruptured the entire slip plane. +Consequently, when it arrived at the geometric limit of the fault, the friction resistance +τf, is infinite by definition. For most small earthquakes it seems likely that the first +case is the applicable one. For larger earthquakes it may be assumed that the second +case is applicable. +ii. This law does not explain how the next earthquake will occur. Following an earthquake, +the entire fault plane that reruptured should, logically, have a shear stress equal to the +dynamic friction multiplied by the effective normal stress i.e., τ = τ d +f = µdσeff. Further, +for the nucleation and propagation of the next earthquake, τ must again increase and +reach the value τ s +f. We talk about a fault plane ‘healing’, but the slip-weakening law +does not allow this. It is thus well-suited to model a single rupture, but not to simulate + +5 +the seismic cycle, where inter-seismic periods and earthquakes succeed one another over +a long period of time. +iii. If we go back to law 4, but µs < µd, we will then have an increase in friction with +the slip, which does not produce instabilities. We then talk of slip-hardening behavior, +which leads to ‘creep’ type behavior. +1.4 +Rate weakening friction law +In order to respond to the problem of the fault plane ’healing’, i.e., to allow the shear value +τ to return to the value τ s +f, Burridge and Knopoff R. Burridge (1967) propose a new model. +They base it on a key observation made in the laboratory: once the plane has slipped from +the critical value δc, the friction becomes a function of the slip rate V : +τf(V ) = (τ s +f − τ d +f ) +V0 +V0 + V + τ d +f +(6) +where V0 corresponds to the characteristic slip velocity. When the slip velocity is much +smaller than V0, the fault’s resistance to slip corresponds to the static friction (µs) multiplied +by the effective normal stress (σeff), i.e., τ s +f. Conversely, when the slip velocity is much +greater than V0, the fault’s resistance to slip corresponds to τ d +f = µdσeff. Therefore, during +an earthquake, the resistance decrease as the slip velocity is large (of the order of 1 m/s). +On the other hand, it rises again quickly as the slip on the fault slows down, when it reaches +loading velocities of the order of a mm/year to cm/year. This, this law can not only model +an earthquake individually, but also model the entire seismic cycle. Burridge et Knopoff +R. Burridge (1967) applied this friction law over a series of connected block-spring systems +used as a proxy for an elastic medium hosting a fault (cf. section 2.1.1). +1.5 +Rate-and-state type friction law +Continuing with the work started by Brace and Byerlee Brace & Byerlee (1966), new exper- +imental protocols have emerged. In particular, researchers wished to explore the effect of the +sudden change in velocity observed in nature, when there is a shift from aseismic velocities +(∼cm/yr) to seismic velocities (∼m/s). Experiments with velocity jumps in the loading of +the system were carried out (figure 3). In his seminal 1998 paper, Chris Marone Marone +(1998) offered an exhaustive review of these works. There are four key observations from +this (figure 4). +• +A sudden change in slip rate first leads to a sudden increase in the coefficient of friction. +This is called the direct effect. +• +A transient adjustment is then seen towards a new, stationary value of the coefficient +of friction. +• +The coefficient of dynamic friction depends on the slip velocity. +• +The coefficient of static friction increases with time when there is no motion between +the two surfaces in contact. +James H. Dieterich was the first person to propose an empirical law that could reproduce +these observations both qualitatively and quantitatively Dieterich (1979a,b). He based this, +notably, on his own friction experiments, with velocity jumps, that involved two ground + +6 +Figure 3. Experiments on friction, by applying velocity jumps, for different types of materials, +published by Dieterich in 1994 Dieterich & Kilgore (1994) +blocks of granodiorite. He also based it on his earlier experiments, demonstrating the co- +efficient of static friction increased with time Rabinowicz (1958). He thus interpreted the +decrease of the coefficient of friction with velocity as an effect of the reduction of the mean +contact time. And so, in his friction law, the coefficient of friction goes from µs to µd over a +distance Dc, which relates the contact time t to the slip velocity V in the following manner: +V = Dc/t. With this, he adopted an approach that was similar to that proposed by E. Ra- +binowicz (cf. section 1.2). The law that he proposed made it possible to bring together the +different coefficients of static and dynamic friction into a single coefficient, which depended +on the slip rate. It was later refined by Ruina (1983), through the introduction of a state +variable θ, which followed a law of evolution. A common way to interpret θ is to relate it +to the lifespan of the asperities present on the surfaces in contact. The law was thus called +the rate-and-state law, due to the existence of this state variable, and the dependence of the +coefficient of friction on the velocity or rate. +A modern form of the rate-and-state law was given by Marone (1998): +τf(V, θ) = +� +µ0 + a log +� V +V0 +� ++ b log +�θV0 +Dc +�� +σeff +(7) +By associating this either with a law called the aging law: +˙θ = 1 − θV +Dc +(8) +or with a state law called slip evolution: +˙θ = −V θ +Dc +log +�V θ +Dc +� +(9) +Here, a > 0 and b are state parameters, of an order of magnitude of ∼ 10−2 , associated, +respectively, with the direct effect and the transient change in the coefficient of friction +(Figure 5). f0 corresponds to the refernce coefficient of friction at the reference velocity V0. +At constant slip velocity, V , the coefficient of friction ad the state variable evolve toward + +.75 +10μm/s +3 μIT/s +10um's +1pm/s +Granite +#60 surface +.70 +15 MPa normal stress +100 μm +.70 +10um/s +1μm/s +10um/s +fum/s +μ +Granite +#60 surface, 1mm gouge +.65 +10 MPa normal stress +100μm +.67 +1μm/s +0.1μm/s +1umis +0.1μm/s +Soda-lime glass +μ +#60 surface +5 MPa normal stress +.62 +100 um +.85 +1μm/s +0.1μm/s +1μm/s +0.1μm/s +Lucite plastic +μ +#60 surface +.70 +2.5 MPa normal stress +100 μm +10μm/s +1μm/s +10μm/s +1μm/s +.025 +Teflon on steel +.020 +polished surface +100 μm +.70 +30MPanormalstress +2μm/s +20um/s +μ +2μm/s +.2μm/s +Wood +.60L +#40 surface +1 MPa normal stress +100 μm7 +day +second +Rate Weakening + +dynamic friction +depends on slip rate +static friction increase with time +when no motion +Additional evolution effect +described mathematically by +an evolving state variable +direct +effect +(Marone, 1998) +(Marone, 1998) +(Marone, 1998) +V1= +0.4 mm/s +V2= +4 mm/s +μd +μs +μ +Figure 4. Experiments on friction. Figures modified as per C. Marone Marone (1998) +Stat +e E +vol +uti +on +Slip Velocity +Friction Coefficient +Time (or Slip) +Time (or Slip) +Figure 5. Schematic illustration of the rate-and-state law + +T +S OI +Quartz gouge +(mm) +15 +25.0 +(ww) + = 3 μm/s +10 s +Load Point Displacement +A +Load Point Displacement +14.5 +s/uw +100 s +24.5 +4 +! +A +14 +A +100 s +- ++ +→ +24.0 +.. +A +1 +0.55 +13.5 +0.66 +0.64 +0.62 +0.54 +1 +6 +三. +105 +104 +. +Ll +L +0 +4 +0 +103 +00 +104 +Hold Time (sec) +L +1000 +. +V (μm/s) +10 +icholz & Engelder 1976 +Johnson &Marone 1997 +100 +Tullis & Weeks 1986 +Beeler et al 1994 +Kilgore et al 1993 +100 +Dieterich 1972 +Dieterich 1981 +Marone 1998 +Dieterich 1978 +1-01 +T +10-2 +: ++ +4 +09'0 ++ +10-3 +0.70, +9.65 +0.60 +5T +S OI +Quartz gouge +(mm) +15 +25.0 +(ww) + = 3 μm/s +10 s +Load Point Displacement +A +Load Point Displacement +14.5 +s/uw +100 s +24.5 +4 +! +A +14 +A +100 s +- ++ +→ +24.0 +.. +A +1 +0.55 +13.5 +0.66 +0.64 +0.62 +0.54 +1 +6 +三. +105 +104 +. +Ll +L +0 +4 +0 +103 +00 +104 +Hold Time (sec) +L +1000 +. +V (μm/s) +10 +icholz & Engelder 1976 +Johnson &Marone 1997 +100 +Tullis & Weeks 1986 +Beeler et al 1994 +Kilgore et al 1993 +100 +Dieterich 1972 +Dieterich 1981 +Marone 1998 +Dieterich 1978 +1-01 +T +10-2 +: ++ +4 +09'0 ++ +10-3 +0.70, +9.65 +0.60 +50.55 +0.54 +0.4mm/s- +4mm/s +Load Point +Quartzgouge +Velocity +13.5 +14 +14.5 +15 +Load Point Displacement +(mm)8 +a stationary value, fss and θss. It is thus possible to rewrite the rate-and-state law as follows: +θss = Dc/V +& +fss = f0 + (a − b) log V +V0 +(10) +Thus, when (a−b) < 0, the coefficient of friction decreases with the increase in slip velocity. +We then speak of a rate-weakening material. If (a−b) > 0 then a rate-strengthening behavior +is obtained. +Today, none of the state laws (equations 8 and 9) reproduce the full set of experimental +data. The slip evolution law does not reproduce the logarithmic time dependence of the +coefficient of static friction (figure 4). If ˙δ = 0, θ does not evolve over time. This is prob- +ably why the models tend to favor the aging law Ampuero & Rubin (2008). However, this +law offers a non-symmetric response according to which a positive (increase) or negative +(decrease) velocity jump is introduced Blanpied et al. (1998); Ampuero & Rubin (2008). +Several modifications were proposed to improve the state law. For example, by introducing +a dependency for the normal stress Linker & Dieterich (1992), by proposing a completely +different evolution of the parameter θ Perrin et al. (1995); Kato & Tullis (2001), or by adding +a dependency to the shear rate Bhattacharya et al. (2015) . However, none of these laws +led to a consensus. On the other hand, other promising modifications made it possible to +come close to observations made in nature (cf section 3.2). Some of those include additional +friction mechanisms that increase friction through dilatancy Segall & Rice (1995); Segall & +Bradley (2012), or lead to a decrease in effective friction through the pressurization of pore +fluids Rice (2006); Schmitt et al. (2011). +2 +Modeling fault behavior: the ‘spring-block slider’ model +In the brittle part of the crust, the deformation is essentially accommodated along faults +in response to the tectonic plate movement in the earth’s crust. Along these faults two +main behaviors are observed: either the fault creeps continuously at a velocity comparable +to the plate velocity (mm/yr to cm/yr), or it remains locked for years, or even centuries, +and slips suddenly in a very short time, of the order of several seconds, thus resulting in an +earthquake. An earthquake of magnitude Mw 4-5 corresponds to an average slip of a few +centimeters, a Mw 7 corresponds to a slip of a few meters, and a Mw 9 to 10 to a slip of 20 +meters. It is thus observed that slips of the order of m/s, cause destructive seismic waves +that propagate in the surrounding medium. A simple analogy to represent the behavior of +faults on the Earth’s surface is the ‘spring-block slider’ model (Figure 6), which is described +in the following section. +2.1 +Modeling the slip on a fault: creep or earthquake +2.1.1 +Block-spring model In the spring-block slider model, the force that pulls on the +spring attached to the block in a constant manner represents the plate motion. The stiffness +constant k of the spring represents the rock’s elastic properties, the weight of the spring, the +compression and basal friction of the block, the friction of the fault plane (Figure 6). There +is therefore competition between the shear force pulling the block, Fspr, and the force of the + +9 +Fspr +Fn (normal force) +k +Ffric +Ft (constant pulling force) +0 +set up +1 +stick +Fspr +k +Ffric = Fs +Displacement δ of point A +Block is +not moving +Stretching of the spring +Fn +Ft +A +2 +slip +Displacement +of the block +Shortening of the spring +k +Fspr +Ffric = Fd +Fn +Ft +A +Figure 6. Spring Block slider model +friction that resists the shear force, Ffric, defined as follows: +Fspr = τ × A = k × x +(11) +Ffric = µ × σeff × A = µ × Fn +(12) +To recall: τ is to the shear stress, A is the contact area, k is the spring’s stiffness coefficient, +σeff is the effective normal stress, and µ is the coefficient of friction. Depending on the law +applicable to µ, for example slip-hardening or slip-weakening, ‘creep’ or ‘earthquakes’ can +be reproduced as observed in nature (cf. section 1.3) . +In the case of faults that produce earthquakes, we speak of stick-slip behavior. That +is, alternating between long periods where the fault does not move but stress accumulates +(stick) and periods where the accumulated stress exceeds the fault’s resistance to slip, which +results in a slip displacement. +2.1.2 +Earthquake and instability condition By applying a slip-weakening law to the +block-spring model, it is therefore possible to reproduce stick-slip behavior and deduce the +instability condition that will lead to a rapid, ‘earthquake’ type slip. +Initially, the spring is pulled over a distance x but the block does not move (phase 1 in +figures 6 and 7). We thus have: +Fspr + Ffric = 0 +(13) +Next, when the shear stress, τ, which is equal to the fault’s resistance to slip, τ s +f = µsσeff, +the block begins to move. Since the block slips in the direction parallel to Fspr, this force +decreases, just as Ffric because we applied a slip-weakening type friction to the model (cf + +10 +EARTHQUAKE +INTERSEISMIC +INTERSEISMIC +Fspr (+) / Ffric (-) +1 +stick +2a slip - acceleration +2b slip-deceleration +Fspr exceeds Ffric here, +so block accelerates. +Now there is an +initeral force too! +Fsprand Ffricare equal +here... but there is still +the initial force! +Ffric is larger than Fspr +here, so block slows +down. +1 +stick +Ffric initially +decreases then +stays constant +forces are +equals here +Fspr which pulls to the +right, is decreasing +δc +Displacement of block +Block not moving +Block not moving +δ +slope =-k +slope = σeff (μs-μd)/ δc +spring is +streching +Block velocity slows +to zero. Spring starts +to strech again, +setting up the next +earthquake +Figure 7. Balance equation of forces for the block-spring model with a slip-weakening friction law +eq. 4). When Fspr exceeds Ffric, the block accelerates (phase 2a. in figure 7). We therefore +add an inertial force to equation 13. +Fspr + Ffric = m¨x +(14) +When the coefficient of friction µ reaches its dynamic value µd, Ffric remains constant, +while Fspr continues to decrease (phase 2b in figure 7). The block finally decelerates. After +it completely stops, phase 1 (the stretching of the spring) resumes. +There is therefore an 1instability’, i.e. an acceleration in slip, when Ffric decreases faster +than Fspr during the slip. The instability condition is, thus, defined through the following +relation, where k, the stiffness of the spring, must be smaller than a critical value kc: +k < kc = +���� +σeff(µs − µd) +δc +���� +(15) +Conversely, creep is produced if k > kc, i.e., if the system is ’rigid’ (a high k) or if the normal +stress is low. +2.1.3 +Representation of a subduction zone. A simple way of representing a subduc- +tion zone, therefore, consists of combining several blocks, connected ot each other through +springs, as proposed by Burridge and Knopoff in 1967 R. Burridge (1967). The aseismic zone +at depth is represented by a block whose basal friction responds to a slip-hardening law, and +the seismogenic zone is represented by a block whose basal friction follows a slip-weakening +law (figure 8a and b). Researchers then observed that for the seismogenic zone, the slip +accumulates in ‘steps’ (figure 8c). This is expressed by jagged variations in the shear stress, +which is accumulated over long periods of time and then released in a few seconds (figure 8d). + +11 +seismogenic zone +creep +Slip-weakening friction +Slip-hardening friction +frictional surface +F (constant pulling force) +A +Time +Position of the slider += cumulative slip +point A +Slip-weakening Model +Inside the seismogenic zone +Slip-hardening Model +Inside the creeping zone +stress accumulation +stress drop +Slip-weakening +Model +Slip-hardening Model +Time +Shear stress +a) +c) +d) +b) +Lorem ipsum +τd s +Figure 8. Modeling of a subduction using the block-spring method. a) chematic representation of +a subduction. b) Conceptual model. c) Accumulation of slip over time. c) State of shear stress over +time. +We then speak of a stress-drop. For the aseismic zone, after going through a plateau, which +corresponds to the time required for the shear stress to reach the block’s value of resistance +to slip, i.e., τ s +d (figure 8d), the slip accumulates continuously and therefore there is indeed +creep (figure 8c). +2.2 +Modeling the seismic cycle +align=left, leftmargin=2em, itemindent=0pt, labelsep=0pt, labelwidth=2em ] +2.2.1 +Shifting to the rate-and-state law As discussed in section 1.3, while the earlier +model makes it possible to reproduce the esential steps that lead to the seismic slip, it +does not allow multiple events to be chained, since µ does not return to its static value µd +(figure 7). On the other hand, the R&S law, with the state variable θ, takes into account +the healing of the fault plane (figure 9). +If we go back to the spring-block slider model and replace the slip weakening friction law +with a rate-and-state friction law, it is possible to derive a new instability condition. In this +second case, during the acceleration phase (2a in figure 9), the slope of Ffric is approximately +equal to σeff(b − a)/Dc. Consequently, for an instability, and potentially an earthquake, to +be generated, we must have the following relation: +k < kc ≈ +���� +σeff(b − a) +Dc +���� +(16) +2.2.2 +Implications for the nucleation size of earthquakes To move from the spring- +block slider model to a slightly more realistic Earth model with elastic behavior, we use + +12 +slope � σeff (a-b)/ Dc +Fspr (+) / Ffric (-) +1 +stick +2a slip - acceleration +2b slip-deceleration +Fspr exceeds Ffric here, +so block accelerates. +Now there is an +initeral force too! +Fsprand Ffricare equal +here... but there is still +the initial force! +Block velocity slows +to zero. Spring starts +to strech again, +setting up the next +earthquake +1 +stick +spring is +streching +Ffric initially +decreases then +stays constant +forces are +equals here +Fspr which pulls to the +right, is decreasing +Dc +Displacement of block +Block not moving +Block not moving +δ +slope =-k +Fault is “healing” +and Ffric increases +as slip slows and +asperity contacts +age +EARTHQUAKE +INTERSEISMIC +INTERSEISMIC +Ffric is larger than Fspr +here, so block slows +down. +Fsprand Ffricare equal +here... but there is still +the initial force! +Figure 9. Assessment of forces for the block-spring model with a rate-weakening friction law (Rate- +and-state law) +elasticity to determine the k value of an elliptical crack: +k = +G +(1 − ν)L +(17) +where G is the shear modulus, ν is the Poisson’s ratio and L is the length of the zone that +slips over the fault plane (figure 10). In this case, the instability occurs when the decrease in +the frictional force is greater than the decrease in elastic force, and equation 16 is rewritten +as: +G +(1 − ν)L < kc ≈ +���� +σeff(b − a) +Dc +���� +(18) +Consequently, the zone that slips must be greater than a critical size Lc in order to become +unstable and generate earthquake nucleation: +L > Lc ≈ +���� +DcG +(1 − ν)σeff(b − a) +���� +(19) +2.2.3 +Continuum model In his seminal 1993 article Rice (1993), J. R. Rice highlights +the importance of moving from ”spring-block slider” models to continuous medium models. +He demonstrated, notably, that “While the equations of Newtonian dynamics are solved +exactly in these Burridge-Knopoff models, it has not been generally acknowledged that the +dynamical solution for rupture along a chain of lumped masses,or a string of concentrated +mass in the continuous limit, bears a presently uncertain relation to dynamical solutions +for rupture along a fault embedded in a surrounding elastic continuum. For example, the + +13 +dx1 +dx2 +fault plane +slipping elliptical patch of +fault length L +elastic rock +with propeties +G and ν +Figure 10. Nucleation model +response of B-K models to an instantaneous change in stress τ along the rupture is an +instantaneous change in the acceleration ∂2δ/∂t2, but there is no instantaneous change in +∂δ/∂t.” This is true, on the other hand, in continuum models. The other major drawback +is “Also, since there is no analogue to energy radiation as seismic waves in the normal +implementation of the B-K models (an exception is the recent work of Knopoff et al. [1992]), +all potential energy lost to the system during a rupture is fully account- able as frictional +work; the same is not true for rupture in a continuum.” +It is therefore essential to highlight, in this text, that while the block-spring model makes +it possible to qualitatively reproduce the phenomena observed in nature, it is essential to +shift to a continuum model if we wish to develop robust numerical models. Interested readers +can consult The mechanics of faulting: from laboratory to real earthquakes Bizzarri & Bhat +(2012). +3 +A more complex physical reality +align=left, leftmargin=2em, itemindent=0pt, labelsep=0pt, labelwidth=2em ] +3.1 +Spatial and temporal variability in the slip mode on faults +Until recently the deformation in fault zones, in the brittle part of the crust, was attributed +either to earthquakes or to the slow, continuous slip during the inter-seismic period (creep) +or post-seismic period. This latter phenomenon is called the afterslip and corresponds to a +logarithmic acceleration in the aseismic slip on the fault, which can be observed after large +earthquakes. However, this paradigm of two ’extreme’ behaviors is being questioned today. +Advances in technology and methodology in the field of geodesy and in seismology have +significantly improved our capacity to measure deformation rates and given us higher res- +olutions. These observations have enabled us to document a large variability in the slip +dynamic in the seismogenic zone (figure 11). Faults may have chiefly seismic behavior, have + +14 ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +++ +creep ? +transition +zone +creep +seismogenic zone +upper plate +Seismic Signals +Geodetic Signals +400 nm/s +5 seconds +Low Frequency Earthquake +Few seconds, M<1.5 +e) +b) +locked patch, slip +from kinematic +inversion +Megathrust Earthquake +several 10s of m in few seconds +50 cm +1 Year +creepmeter +Creep, upper segment +~ few cm in 1 year +1 Year +a) +30 cm +600 nm/s +10 seconds +Megathrust Earthquake +Few seconds to mins, M<9.5 +d) +10 Days +c) +Deep Slow Slip Svent +~ 0.5 mm in 1 day +5 mm +50 nm/s +200 seconds +Very Low Frequency +earthquake +Tens of seconds, M<4 +g) +600 nm/s +200 seconds +Non Volcanic Tremors +Seconds to hours, M<1.5 +f) +stable creep +Generated by +microcraks? +swarm of low +frequency +earthquakes? +Figure 1.11. Représentation schématique d’une zone de subduction et de la distribution des sources potentiellement responsables des phéno- +mènes observés. GEODESIE : a) fluage enregistré par un extensomètre, b) glissement durant un séisme majeur , c) slow slip events (SSE). +SISMOLOGIE : sismogrammes enregistrant (d) séismes, (e) Low Frequency Earthquakes, (f) non volcanic tremors, et (g) Very Low Frequency +Earthquakes . Les patchs rouges représentent les aspérités “bloquées" pendant la période intersismique. Les isocontours mauves représentent +le glissement cosismique qui aurait rompu plusieurs patchs adjacents. Les données en a) et b): [THO 14] ; c), d), e), f) : [PEN 10] +. + +wFreguency +rthguake(LFE) +-EarthquakeVeryLow +Frequency +Event (VLF)Tremor15 +a slow, stable slip Thomas et al. (2014a) or a transient slip Rousset et al. (2016). In ad- +dition to this, one of the most significant discoveries in the last decade has been revealing +the existence of ’slow earthquakes’ (cf. Chapter 7). These encompass several phenomena. +Slow slip events rupture the fault very slowly over several hours or even days, at velocities +that are higher than the inter-seismic creep (cm/yr), but slower than earthquakes, such that +no detectable seismic waves are radiated Dragert et al. (2001). They are generally (though +not always) accompanied by weak seismic signals of a long duration (a few minutes to a +few weeks) called non volcanic tremors Obara (2002). Low frequency earthquakes, with a +duration close to a second, and Very low frequency earthquakes, which can last a hundred +seconds, are commonly observed within non volcanic tremors Ide et al. (2007); Ito et al. +(2007). As a result, it is known today that slip velocities on faults cover a continuum going +from a millimeter per year to a meter per second Peng & Gomberg (2010). This is therefore +an essential parameter to take into consideration when modeling active faults. However, the +physics behind the processes that govern this behavior is still unknown and is the subject +of much active debate in the community. +In addition to the large range of deformation velocities there is a spatial and temporal +variability in the slip mode. Contrary to what the schematic representation of figure 11 might +suggest, the phenomena described here are not restricted to a specific depth. On some faults +creep may be recorded over the entire seismogenic zone i.e. from the surface up to the +maximum depth where earthquakes are observed Titus et al. (2006); Thomas et al. (2014a). +Further, while slow earthquakes were first located beyond the seismogenic zone Obara (2002); +Ide et al. (2007), non volcanic tremors and slow slip events have recently been observed at +epths of less than 10 km, as well as in the sub-surface Ito & Obara (2006); Outerbridge +et al. (2010). Moreoever, geodetic data has shown that the seismic or aseismic behavior is +not necessarily stable over time, and that the same zone may creep and slide seismically +Johnson et al. (2012); Thomas et al. (2017a). These observations lead to two hypotheses. (1) +These different phenomena can occur under varied pressure/temperature conditions and/or +result from various deformation mechanisms. (2) They correspond to particular mechanical +and rheological properties, but which vary over time. Consequently, they also vary over +space, depending on what seismic cycle phase the observed site is undergoing. +3.2 +Additional mechanisms that can come into play during earthquakes +The standard formulation of the rate-and-state law, (section 1.5), allows a numerical re- +production of a large number of the phenomena discussed above. However, this formulation +was based on slip velocity experiments ranging from 10−9 to 10−3 m/s. While comparable +to aseismic velocities (10−10 to 10−9 m/s), they are still slow when compared to seismic +velocities (∼ 1 m/s). There is increasing experimental and theoretical proof that larger slip +velocities and quantity of slip also come into play Lapusta & Barbot (2012). This has the ef- +fect of drastically reducing the dynamic friction. Wibberley and co-authors Wibberley et al. +(2008) have compiled laboratory values for different kinds of rocks and at different loading +velocities (figure 12). +The lack of experimental data on the properties of friction that are applicable to earth- +quakes is due to the difficulty of carrying out experiments in conditions similar to earth- +quakes. A laboratory experiment that would reproduce the conditions that exist during + +16 +Typical slow-slip values +Earthquakes +Plate motion +mm/yr +cm/yr +m/s +Figure 12. Dependence of the coefficient of dynamic friction, in a continuous regime, on the slip +velocity. Figure modified as per Wibberley et al. (2008). +seismic slip would simultaneously involve high slip rates (1-10 m/s), with large displace- +ments (0.1-20 m), a resulting effective normal stress (50-200 MPa), high pore pressure (0.4 +to 1 times the normal stress) and high temperature (ambient temperatures of 100 to 300◦C, +but potentially as high as 1500◦ C in the slip zone). Although considerable progress has +been made over the last decade, there is as yet no device that is capable of simultaneously +responding to all these requirement. It is therefore necessary to compromise on one or more +factors. Tullis and Schubert highlighted this difficulty and proposed a complete review of +the processes that could lead to substantial reductions in the friction coefficient with respect +to its typical experimental value of 0.6 Tullis & Schubert (2015). The proposed mechanisms +include: +• +dynamic reduction in the normal stress or loss of contact due to the vibrations perpen- +dicular to the interface, +• +dynamic reduction in the normal stress due to the contrast in elastic properties, or +permeability, on either side of the fault, +• +acoustic fluidization, +• +elasto-hydrodynamic lubrication, +• +thermal pressurization of pore fluids, +• +pressurization of pore fluids induced by the degradation of minerals, +• +local heating/melting of the point of contact between the asperities, +• +lubrication of the fault through fusion, in response to frictional processes, +• +lubrication of the fault through the creation of a thixotropic silica gel, +• +superplastic deformation of fine grains. +These highlight the difficulty of proving which mechanism is responsible for the observed ex- +perimental behavior and to design experiments that can clearly prove or refute a mechanism +proposed in theory. Nonetheless, since it is likely that one or more of these processes is acti- +vated at high slip rates, the rate-and-state law described in section 1.5 does not adequately + +1.0 +0.8 +田 +田 +田 +M +V +田 +芒 +V +田 +0.6 +0 +Friction coefficie +G +AV +Peridotite 20 MPa [Di Toro et al.2006b] +Gabbro 15.5 MPa [Nielsen et al. 2008] +G +Tonalite 20 MPa [Di Toro et al. 2006b] +0 +Clay-rich gouge 0.6 MPa [Mizoguchi et al. 2007] +Granite 2 MPa [Dieterich 1978] +0.4 +A +田 +Aplite 5 MPa [Di Toro et al. unpublished] +Westerly granite 5 MPa [Di Toro et al.2004] +Feldspar 5 MPa [Di Toro et al. unpublished] +Westerly granite 5 MPa [Di Toro et al. unpublished] +G +Feldspar1.3 MPa [Di Toro et al. unpublished] +会 +Serpentinite 1.5 MPa [Hirose and Bystricky 2007] +Carrara marble 12-15 MPa [Han et al. 2007; Han et al. in prep] +V +Serpentinite 2.5 MPa [Hirose and Bystricky 2007] +Carrara marble <10 MPa [Han et al. 2007; Han et al. in prep] +Serpentinite 24.5 MPa [Hirose and Bystricky 2007] +Carrara marble 1.25 MPa [Di Toro unpublished] +V +0.2 +Serpentinite 6.5 MPa [Hirose and Bystricky 2007] +Calcite gouge [Shimamoto and Logan 1981] +Serpentinite 15.5 MPa [Hirose and Bystricky 2007] +Calcite gouge 100 MPa [Morrow et al. 2000] +Serpentinite 25 MPa bare surface [Reinen et al. 1992] +Dolomite marble 75 MPa[Weeks and Tullis 1985] +0 +Serpentinite 25 MPa gouge-present[Reinen et al. 1992] +Dolomite marble 7.2 MPa [Han et al. in prep] +Quartz sandstone 18.7 MPa [Dieterich 1978] +口 +Dolomite marble 12.2 MPa [Han et al. in prep] +Novaculite 5 MPa [Di Toro et al. 2004] +Dolomite gouge [Shimamoto and Logan 1981] +0.0 +1 +-6 +1 +1 +-11 +-10 +6- +-8 +-7 +-5 +-4 +-3 +-2 +-1 +0 +Log slip-rate (m s-1)17 +reproduce this strong fall in the coefficient of dynamic friction. Indeed, for seismic velocities +(∼ 1 m/s) is a typical value for (a − b) equal to −0.005, we obtain a µd of ∼ 0.54. Further, +based on laboratory experiments, the effectve µd, i.e., τ/σeff, can reach very low values (0 +to 0.2) during co-seismic slip. This observation has many implications for our understanding +of the mechanism of earthquakes: on the amplitude of the stress drop, on the propensity of +earthquakes to propagate in pulse form, on the amplitude of ground movements, and on the +orientation of stresses in the crust. N. Lapusta and S. Barbot propose two ways of modifying +the rate-and-state law to take into account these additional weakening mechanisms Lapusta +& Barbot (2012). Interested readers may refer to their publication for more details. +3.3 +Going beyond the elastic Earth model +Many ground studied, geophysical observations, and laboratory experiments have high- +lighted the strong coupling that exists between the main rupture plane and the surrounding +medium. . The faults zones are not made up only of a major plane where the majority of +slip occurs, but also make up a complex group, surrounded by a zone where surrounding +rock is fractured intensively (figure 13). Seismic ruptures result in damage around the faults +with an exponential decrease in the density of microfractures perpendicular to the main +slip plane Anders & Wiltschko (1994); Mitchell & Faulkner (2009). The damage modifies +the microstructure and changes the elastic properties of the rocks at the level of the fault +breccia and in the adjacent medium Walsh (1965a,b); Faulkner et al. (2006). These changes, +in return, modify the extension an dynamic of the rupture as well as the radiation of seismic +waves Thomas et al. (2017b). They also influence seismic processes during the post-seismic +period, such as aftershocks, with the minimum size of the nucleation zone depending chiefly +on the elastic modulus Rubin & Ampuero (2005). In their experimental study, Gratier et al. +(2014) have also demonstrated that the co-seismic damage would promote aseismic slip +through pressure-dissolution, thus explaining the afterslip recorded after large earthquakes. +Co-seismic damage also increases permeability (figure 13e), which results in a variation +in the fluid pressure Sibson (1994) that modifies the fault’s resistance to slip. Geophysical +observations suggest that this effect is transient (figure 13d), because a gradual and partial +recovery of the elastic properties after the earthquake has been recorded Hiramatsu et al. +(2005); Froment et al. (2014). This evolution is probably related to the healing of microfrac- +tures and faults through the precipitation of dissolved substances, products of alteration +and/or the development of clayey minerals Mitchell & Faulkner (2008). In their model, den +Hartog & Spiers (2014) propose that the compaction through pressure-dissolution leads in +turn to the recovery of seismogenic behavior. +Moreover, several studies have demonstrated the influence of the properties of the sur- +rounding rock on the behavior of faults. Audet and co-authors have shown a direct rela- +tionship between the physical properties of the interlocking plate in the subduction zone +and the recurrence of slow earthquakes Audet & Burgmann (2014). In my microstructure +study of Taiwan’s longitudinal valley fault, Thomas and co-authors were able to demonstrate +the aseismic behavior of the fault was controlled by inherited microstructure Thomas et al. +(2014b). Perrin et al. (2016) looked at the influence of the ’maturity’ of the faults on the +accumulation of slip. A study of 27 earthquakes concluded that the more damage the fault +presents (mature fault), the greater the quantity of slip during an earthquake. + +18 +Fluid flow? +Pore Pressure? +Alteration ? +Elastic properties? +Temperature? +Healing? +Permeability? +creep ? +transition +zone +creep +locked zone +0 +50 +100 +150 +200 +2 +5 +10 +15 +20 +Distance from the fault core (m) +Log cracks Density (#/mm) +Caleta Coloso Fault +Blanca fault +Background +microfracture +Damage Density +a) +at peak stress of cycle +at 60 % failure stress +10-17 +10-18 +10-19 +Permeability (m ) +2 +Permeability +changes with damage density +Microfracture density (#/mm) +2 +4 +6 +10 +12 +14 +16 +8 +e) +70 +66 +16 +8 +0 +58 +54 +62 +Number of cycles +Poisson’s ratio +0.25 +0.50 +0.45 +0.40 +0.35 +0.30 +Microfracture +density (#/mm) +Young’s modulus (GPa) +2 +4 +6 +1 +3 +7 +9 +8 +5 +Elastic Properties +changes with damage density +40 MPa +60 MPa +80 MPa +100 MPa +c) +Temperature +Variation of (a-b) +Temperature (°C) +0 +100 +200 +300 +400 +500 +Illite gouge +(a-b) +-0.02 -0.01 0.00 0.01 0.02 0.03 +b) +Damage Zone +Alteration +Fault core +Fault breccia +250 +300 +350 +Julian day in 2008 +-10 +-8 +-6 +-4 +-2 +0 +dV/V (%) +Healing +Temporal changes in seismic velocity +d) +SEISMIC +ASEISMIC +Westerly granite +Gofar transform fault +Westerly granite +Figure 1.13. Représentation schématique, du point de vue de la mécanique, d’une zone de subduction. L’interface entre la plaque plongeante et la plaque +chevauchante est une zone complexe qui comprend une brèche de faille, des plans de glissement principaux, et une zone d’endommagement. La deforma- +tion dépend de plusieurs paramètres qui ont leur propres évolutions temporelles durant le cycle sismique. (a) L’endommagement décroit de façon exponen- +tielle avec la distance à la faille [MIT 09]. (b) Les paramètres de friction (a − b) dépendent de la température [HAR 12]. (c) Les propriétés élastiques +varient avec l’endommagement [FAU 06]. (d) La vitesse des ondes sismiques changent après un séisme [FRO 14]. Le trait pointillé marque la date d’un +séisme de Mw 6.0. (e) La perméabilité évolue en fonction de la densité de microfracture [MIT 12]. +. + +19 +4 +Transition towards a new generation of models +The usual way of looking at the fault restricts the deformation in the brittle part of the +crust to slip along the interface (fault plane), loaded with creep at depth, whose behavior +is controlled by its frictional properties Scholz (1998). According to these properties, when +the resistance threshold is exceeded, the stress accumulated when the fault is locked is +released through seismic slip or creep, or again during slow earthquakes. Further, as the +previously cited studies have highlighted, while the behavior of the fault zones is intrinsically +related to the properties of the main slip plane, it also depends on the properties of the +surrounding rock. In parallel, the displacement on the faults induces a modification of the +physical properties of the surrounding medium. these observations suggest the existence of a +second ‘cycle’ where the properties of the fault zone evolve with respect to the slip dynamic, +which in turn influences the deformation mode. +However, the majority of models used today do not take this complex feedback into ac- +count. By attributing constant properties (pressure, temperature, petrology, microstructure) +that do not evolve with deformation, we neglect to take into account how seismic/aseismic +fault behavior is impacted by temporal variations of the physical properties of the volume +and the interface. It is thus useful to develop a new generation of models that take into +account spatial-temporal evolution of physical properties in fault zones. New models are +being developed and have already shown the importance of these interactions from a seismic +point of view Thomas et al. 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On the history of the dry friction law, Mechanics of solids, 48(4), 364–369. + diff --git a/htAzT4oBgHgl3EQfo_1S/content/tmp_files/load_file.txt b/htAzT4oBgHgl3EQfo_1S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9112196d514b6f33b8319b232ea359970b169f45 --- /dev/null +++ b/htAzT4oBgHgl3EQfo_1S/content/tmp_files/load_file.txt @@ -0,0 +1,1058 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf,len=1057 +page_content='1 Friction Laws and numerical modeling of the seismic cycle To appear as a chapter in “The Seismic Cycle: From Observation to Modeling edited by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Rolandone” Marion Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Thomas1 and Harsha S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Bhat2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Institut des Sciences de la Terre Paris, Sorbonne Universit´e, CNRS-UMR 7193, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Laboratoire de G´eologie, ´Ecole Normale Sup´erieure, CNRS-UMR 8538, PSL Research Uni- versity, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1 Friction Laws 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1 Historical notions about friction Friction is resistance to motion that appears when two surfaces in contact slide against one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Generally speaking, the concept of ‘friction’ describes the dissipation of energy that occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Most phenomena associated with sliding friction can be understood from observations made by Leonardo da Vinci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' He was the first to note that, based on his experiments, friction is proportional to 1/4th of the applied pressure and that it is independent of the area of contact between two active surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This latter observation was inspired by the fact that the resistance to sliding of a coil of rope is the same as for a stretched piece of rope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Almost two centuries later, in the 18th century, Guillaume Amontons and Charles- Augustin de Coulomb, carried out rigorous experiments on friction, with the aim of ob- taining quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The collective work by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' da Vinci, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Amontons and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' de Coulomb led to the two fundamental ’laws’ of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' These statements, simple and still valid, are widely applicable: the friction force acting between two sliding surfaces is proportional to the load pressing the surfaces together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' That is, these two forces have a constant ratio, often called the coefficient of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' the sliding force is independent of the apparent area of contact between the two surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The discoveries that followed (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Chapter 1), led researchers to revisit these laboratory experiments in order to better understand earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In 1966, in a now-famous paper, Brace and Byerlee showed that the creation of new fractures was not the only model that could explain the existence of seismic faults Brace & Byerlee (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In their experimental protocol, they pre-cut a rock sample and loaded its extremities, while also applying confining pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' They observed that the sliding between the two pieces of rock was not continuous, but a jerky motion with accelerations and decelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This was the origin of the theory, which is widely accepted today, that earthquakes are governed by frictional forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 From static friction to dynamic friction If we go back to the fundamental laws of friction stated by Amonton-Coulomb, they are mathematically expressed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The frictional force Ffric = τA is independent of the contact area A (τ being the shear stress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ffric is proportional to the applied normal force arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='01605v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='geo-ph] 3 Jan 2023 2 Fn = σeffA through the constant µ (σeff corresponds to the effective normal stress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We thus have: µ = Ffric Fn = τ σeff (1) Let us now consider an object of mass M placed on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The force Fn = Mg is, therefore, normal to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We apply a tangential force Ft parallel to the surface of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' If the object is initially at rest, a motion may be produced if a force Ft, greater than Ffric, is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In this case, the coefficient µs is called the coefficient of static friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ffric = Fs = µsFn (2) Now, if the object is displaced at a finite velocity over the surface, it has been experimentally found that the frictional force is also proportional to the normal force, through the coefficient µd, called the coefficient of dynamic friction: Ffric = Fd = µdFn (3) Early experiments showed that the coefficient of static friction is different from the coeffi- cient of dynamic friction Rabinowicz (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Static friction has the property of increasing logarithmically with time, and dynamic friction depends on the velocity V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' From the classic work carried out by Kostrov Kostrov (1964, 1966) and Eshelby Eshelby (1969), it soon became clear that friction also played a fundamental role in the initiation, rupture development and ‘healing’ of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The classic Amonton-Coulomb model, however, led to an impasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Among other physical problems, it postulated the hypothesis of an instan- taneous modification of the coefficient of friction, from its static value to its dynamic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This brings in singularities (infinite stresses) at the rupture front (red model in figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This model lacks a scale of length that makes it possible to define a finite quantity of energy released at the rupture front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' There are two possible options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' One consists of defining the characteristic quantity of slip (between the two surfaces) required to move from static friction to dynamic friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The other consists of introducing a characteristic time in which friction decreases from µs to µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In this second case, a scale of characteristic length emerges when the characteristic time is related to the slip velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' For example: to explain his experiments on friction, Rabinowicz Rabinowicz (1958) introduced the concept of a ”critical distance” dc during which the gap between the static friction and the dynamic friction is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' He related this critical distance to the velocity, V = Dc/tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Here tw is called weakening time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In general, the laws called weakening friction laws were thus developed to reproduce seismic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We speak of weakening because the friction reduces with the slip (or rate of slip) and these laws can thereby produce instabilities Bocquet (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Zhuravlev (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Romanet (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This ingredient is required to anticipate seismic velocities (m/s) in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We will now present the most used models in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3 Slip weakening friction law In fracture mechanics, the model where friction weakens with distance, also known as the cohesive zone model, postulates that: 3 position Stress c) position Slip b) x2 x1 μd slip a) Friction A B δc μs ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' position cohesive zone crack x2 x1 ∞ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Comparison between the rupture model hypothesizing linear elasticity (red curve) and the cohesive zone model (dotted blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' a) Coefficient of friction in terms of the quantity of slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' b) Quantity of slip in terms of the position along the fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The point x1 is in the position A on the friction curve and the point x2 is at position B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' c) Stress field close to the rupture front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' the rupture process, which causes the shift from static friction to dynamic friction, is confined to the fracture plane, inelastic deformation begins when the stresses on the rupture front reach a certain critical level, we reach the value of the coefficient of dynamic friction when the displacement on the fracture plane exceeds a critical value δc Leonov & Panasyuk (1959);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Barenblatt (1959);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Dugdale (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This law was introduced in the context of a study of tension fractures, in order to solve the problem of singularities coming up (infinite stresses) on the rupture front (blue model in figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The slip weakening friction law was introduced by Ida Ida (1972) and Andrews Andrews (1976) to model dynamic ruptures for 2D models, and by Day Day (1982) for 3D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This is analogous to the cohesive zone model, but for mode II fractures, that is, for shear fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In this law, the slip is zero until the shear stress τ reaches a maximum value (elasticity limit) that will be denoted by τ s f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Once this stress is attained, the slip starts and the resistance to the sliding τf decreases linearly until the value τ d f , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', when the plane has 4 Slip Friction μs μd δc Slip Velocity Friction μs μd a) b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Schematic illustration of (a) the slip weakening friction law, (b) the velocity weakening friction law slipped with a critical value δc: τf(δ) = � � � (τ s f − τ d f ) � 1 − δ δc � + τ d f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' δ < δc τ d f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' δ > δc (4) If this law is combined with the Amonton-Coulomb law (equation 1), we have: τf(δ) = � � � � (µs − µd) � 1 − δ δc � + µd � σeff ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' δ < δc µdσeff ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' δ > δc (5) where µd < µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In their article, Palmer and Rice Palmer & Rice (1973) presented a law that is very close to this for which they could derive a complete analytical solution for the rupture front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' They showed that this law made it possible to regularize the numerical model by distributing the stresses and the slip over a distance controlled by the length scale in the friction law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A few nuanced but important points with respect to the slip weakening law: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This friction law describes the start and growth of a seismic rupture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The more the fault slips, the weaker its resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' If the shear stress on the fault, τ, is uniform, then this law implies that the fault will continue to slip indefinitely until τ < τf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This does not match the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' There are therefore two possibilities: either τ is heterogeneous along the the fault due to its geometric complexity (branches, non-linear plane, fault jump etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=') or related to past earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The second possibility, since faults have finite length, is that the rupture stoped because the earthquake ruptured the entire slip plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Consequently, when it arrived at the geometric limit of the fault, the friction resistance τf, is infinite by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' For most small earthquakes it seems likely that the first case is the applicable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' For larger earthquakes it may be assumed that the second case is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This law does not explain how the next earthquake will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Following an earthquake, the entire fault plane that reruptured should, logically, have a shear stress equal to the dynamic friction multiplied by the effective normal stress i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', τ = τ d f = µdσeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Further, for the nucleation and propagation of the next earthquake, τ must again increase and reach the value τ s f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We talk about a fault plane ‘healing’, but the slip-weakening law does not allow this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' It is thus well-suited to model a single rupture, but not to simulate 5 the seismic cycle, where inter-seismic periods and earthquakes succeed one another over a long period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' If we go back to law 4, but µs < µd, we will then have an increase in friction with the slip, which does not produce instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We then talk of slip-hardening behavior, which leads to ‘creep’ type behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='4 Rate weakening friction law In order to respond to the problem of the fault plane ’healing’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', to allow the shear value τ to return to the value τ s f, Burridge and Knopoff R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Burridge (1967) propose a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' They base it on a key observation made in the laboratory: once the plane has slipped from the critical value δc, the friction becomes a function of the slip rate V : τf(V ) = (τ s f − τ d f ) V0 V0 + V + τ d f (6) where V0 corresponds to the characteristic slip velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' When the slip velocity is much smaller than V0, the fault’s resistance to slip corresponds to the static friction (µs) multiplied by the effective normal stress (σeff), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', τ s f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Conversely, when the slip velocity is much greater than V0, the fault’s resistance to slip corresponds to τ d f = µdσeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Therefore, during an earthquake, the resistance decrease as the slip velocity is large (of the order of 1 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' On the other hand, it rises again quickly as the slip on the fault slows down, when it reaches loading velocities of the order of a mm/year to cm/year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This, this law can not only model an earthquake individually, but also model the entire seismic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Burridge et Knopoff R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Burridge (1967) applied this friction law over a series of connected block-spring systems used as a proxy for an elastic medium hosting a fault (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 Rate-and-state type friction law Continuing with the work started by Brace and Byerlee Brace & Byerlee (1966), new exper- imental protocols have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In particular, researchers wished to explore the effect of the sudden change in velocity observed in nature, when there is a shift from aseismic velocities (∼cm/yr) to seismic velocities (∼m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Experiments with velocity jumps in the loading of the system were carried out (figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In his seminal 1998 paper, Chris Marone Marone (1998) offered an exhaustive review of these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' There are four key observations from this (figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A sudden change in slip rate first leads to a sudden increase in the coefficient of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This is called the direct effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A transient adjustment is then seen towards a new, stationary value of the coefficient of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The coefficient of dynamic friction depends on the slip velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The coefficient of static friction increases with time when there is no motion between the two surfaces in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' James H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Dieterich was the first person to propose an empirical law that could reproduce these observations both qualitatively and quantitatively Dieterich (1979a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' He based this, notably, on his own friction experiments, with velocity jumps, that involved two ground 6 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Experiments on friction, by applying velocity jumps, for different types of materials, published by Dieterich in 1994 Dieterich & Kilgore (1994) blocks of granodiorite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' He also based it on his earlier experiments, demonstrating the co- efficient of static friction increased with time Rabinowicz (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' He thus interpreted the decrease of the coefficient of friction with velocity as an effect of the reduction of the mean contact time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' And so, in his friction law, the coefficient of friction goes from µs to µd over a distance Dc, which relates the contact time t to the slip velocity V in the following manner: V = Dc/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' With this, he adopted an approach that was similar to that proposed by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ra- binowicz (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The law that he proposed made it possible to bring together the different coefficients of static and dynamic friction into a single coefficient, which depended on the slip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' It was later refined by Ruina (1983), through the introduction of a state variable θ, which followed a law of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A common way to interpret θ is to relate it to the lifespan of the asperities present on the surfaces in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The law was thus called the rate-and-state law, due to the existence of this state variable, and the dependence of the coefficient of friction on the velocity or rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A modern form of the rate-and-state law was given by Marone (1998): τf(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' θ) = � µ0 + a log � V V0 � + b log �θV0 Dc �� σeff (7) By associating this either with a law called the aging law: ˙θ = 1 − θV Dc (8) or with a state law called slip evolution: ˙θ = −V θ Dc log �V θ Dc � (9) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' a > 0 and b are state parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' of an order of magnitude of ∼ 10−2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' associated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' with the direct effect and the transient change in the coefficient of friction (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' f0 corresponds to the refernce coefficient of friction at the reference velocity V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' At constant slip velocity, V , the coefficient of friction ad the state variable evolve toward .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content="75 10μm/s 3 μIT/s 10um's 1pm/s Granite #60 surface ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='70 15 MPa normal stress 100 μm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='70 10um/s 1μm/s 10um/s fum/s μ Granite #60 surface, 1mm gouge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='65 10 MPa normal stress 100μm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='67 1μm/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1μm/s 1umis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1μm/s Soda-lime glass μ #60 surface 5 MPa normal stress .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='62 100 um .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='85 1μm/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1μm/s 1μm/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1μm/s Lucite plastic μ #60 surface .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa normal stress 100 μm 10μm/s 1μm/s 10μm/s 1μm/s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='025 Teflon on steel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='020 polished surface 100 μm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='70 30MPanormalstress 2μm/s 20um/s μ 2μm/s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2μm/s Wood .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='60L #40 surface 1 MPa normal stress 100 μm7 day second Rate Weakening dynamic friction depends on slip rate static friction increase with time when no motion Additional evolution effect described mathematically by an evolving state variable direct effect (Marone, 1998) (Marone, 1998) (Marone, 1998) V1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='4 mm/s V2= 4 mm/s μd μs μ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Experiments on friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Figures modified as per C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Marone Marone (1998) Stat e E vol uti on Slip Velocity Friction Coefficient Time (or Slip) Time (or Slip) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Schematic illustration of the rate-and-state law T S OI Quartz gouge (mm) 15 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0 (ww) = 3 μm/s 10 s Load Point Displacement A Load Point Displacement 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 s/uw 100 s 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A 14 A 100 s ++ → 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='. A 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='55 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='54 1 6 三.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 105 104 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ll L 0 4 0 103 00 104 Hold Time (sec) L 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=" V (μm/s) 10 icholz & Engelder 1976 Johnson &Marone 1997 100 Tullis & Weeks 1986 Beeler et al 1994 Kilgore et al 1993 100 Dieterich 1972 Dieterich 1981 Marone 1998 Dieterich 1978 1-01 T 10-2 : + 4 09'0 + 10-3 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='70, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='60 5T S OI Quartz gouge (mm) 15 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0 (ww) = 3 μm/s 10 s Load Point Displacement A Load Point Displacement 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 s/uw 100 s 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A 14 A 100 s ++ → 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='. A 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='55 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='54 1 6 三.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 105 104 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ll L 0 4 0 103 00 104 Hold Time (sec) L 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=" V (μm/s) 10 icholz & Engelder 1976 Johnson &Marone 1997 100 Tullis & Weeks 1986 Beeler et al 1994 Kilgore et al 1993 100 Dieterich 1972 Dieterich 1981 Marone 1998 Dieterich 1978 1-01 T 10-2 : + 4 09'0 + 10-3 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='70, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='60 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='4mm/s- 4mm/s Load Point Quartzgouge Velocity 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 15 Load Point Displacement (mm)8 a stationary value, fss and θss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' It is thus possible to rewrite the rate-and-state law as follows: θss = Dc/V & fss = f0 + (a − b) log V V0 (10) Thus, when (a−b) < 0, the coefficient of friction decreases with the increase in slip velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We then speak of a rate-weakening material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' If (a−b) > 0 then a rate-strengthening behavior is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Today, none of the state laws (equations 8 and 9) reproduce the full set of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The slip evolution law does not reproduce the logarithmic time dependence of the coefficient of static friction (figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' If ˙δ = 0, θ does not evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This is prob- ably why the models tend to favor the aging law Ampuero & Rubin (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' However, this law offers a non-symmetric response according to which a positive (increase) or negative (decrease) velocity jump is introduced Blanpied et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ampuero & Rubin (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Several modifications were proposed to improve the state law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' For example, by introducing a dependency for the normal stress Linker & Dieterich (1992), by proposing a completely different evolution of the parameter θ Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Kato & Tullis (2001), or by adding a dependency to the shear rate Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2015) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' However, none of these laws led to a consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' On the other hand, other promising modifications made it possible to come close to observations made in nature (cf section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Some of those include additional friction mechanisms that increase friction through dilatancy Segall & Rice (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Segall & Bradley (2012), or lead to a decrease in effective friction through the pressurization of pore fluids Rice (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Schmitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2 Modeling fault behavior: the ‘spring-block slider’ model In the brittle part of the crust, the deformation is essentially accommodated along faults in response to the tectonic plate movement in the earth’s crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Along these faults two main behaviors are observed: either the fault creeps continuously at a velocity comparable to the plate velocity (mm/yr to cm/yr), or it remains locked for years, or even centuries, and slips suddenly in a very short time, of the order of several seconds, thus resulting in an earthquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' An earthquake of magnitude Mw 4-5 corresponds to an average slip of a few centimeters, a Mw 7 corresponds to a slip of a few meters, and a Mw 9 to 10 to a slip of 20 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' It is thus observed that slips of the order of m/s, cause destructive seismic waves that propagate in the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A simple analogy to represent the behavior of faults on the Earth’s surface is the ‘spring-block slider’ model (Figure 6), which is described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1 Modeling the slip on a fault: creep or earthquake 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1 Block-spring model In the spring-block slider model, the force that pulls on the spring attached to the block in a constant manner represents the plate motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The stiffness constant k of the spring represents the rock’s elastic properties, the weight of the spring, the compression and basal friction of the block, the friction of the fault plane (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' There is therefore competition between the shear force pulling the block, Fspr, and the force of the 9 Fspr Fn (normal force) k Ffric Ft (constant pulling force) 0 set up 1 stick Fspr k Ffric = Fs Displacement δ of point A Block is not moving Stretching of the spring Fn Ft A 2 slip Displacement of the block Shortening of the spring k Fspr Ffric = Fd Fn Ft A Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Spring Block slider model friction that resists the shear force, Ffric, defined as follows: Fspr = τ × A = k × x (11) Ffric = µ × σeff × A = µ × Fn (12) To recall: τ is to the shear stress, A is the contact area, k is the spring’s stiffness coefficient, σeff is the effective normal stress, and µ is the coefficient of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Depending on the law applicable to µ, for example slip-hardening or slip-weakening, ‘creep’ or ‘earthquakes’ can be reproduced as observed in nature (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In the case of faults that produce earthquakes, we speak of stick-slip behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' That is, alternating between long periods where the fault does not move but stress accumulates (stick) and periods where the accumulated stress exceeds the fault’s resistance to slip, which results in a slip displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 Earthquake and instability condition By applying a slip-weakening law to the block-spring model, it is therefore possible to reproduce stick-slip behavior and deduce the instability condition that will lead to a rapid, ‘earthquake’ type slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Initially, the spring is pulled over a distance x but the block does not move (phase 1 in figures 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We thus have: Fspr + Ffric = 0 (13) Next, when the shear stress, τ, which is equal to the fault’s resistance to slip, τ s f = µsσeff, the block begins to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Since the block slips in the direction parallel to Fspr, this force decreases, just as Ffric because we applied a slip-weakening type friction to the model (cf 10 EARTHQUAKE INTERSEISMIC INTERSEISMIC Fspr (+) / Ffric (-) 1 stick 2a slip - acceleration 2b slip-deceleration Fspr exceeds Ffric here, so block accelerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Now there is an initeral force too!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Fsprand Ffricare equal here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' but there is still the initial force!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ffric is larger than Fspr here, so block slows down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1 stick Ffric initially decreases then stays constant forces are equals here Fspr which pulls to the right, is decreasing δc Displacement of block Block not moving Block not moving δ slope =-k slope = σeff (μs-μd)/ δc spring is streching Block velocity slows to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Spring starts to strech again, setting up the next earthquake Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Balance equation of forces for the block-spring model with a slip-weakening friction law eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' When Fspr exceeds Ffric, the block accelerates (phase 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' in figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We therefore add an inertial force to equation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Fspr + Ffric = m¨x (14) When the coefficient of friction µ reaches its dynamic value µd, Ffric remains constant, while Fspr continues to decrease (phase 2b in figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The block finally decelerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' After it completely stops, phase 1 (the stretching of the spring) resumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' There is therefore an 1instability’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' an acceleration in slip, when Ffric decreases faster than Fspr during the slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The instability condition is, thus, defined through the following relation, where k, the stiffness of the spring, must be smaller than a critical value kc: k < kc = ���� σeff(µs − µd) δc ���� (15) Conversely, creep is produced if k > kc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', if the system is ’rigid’ (a high k) or if the normal stress is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3 Representation of a subduction zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A simple way of representing a subduc- tion zone, therefore, consists of combining several blocks, connected ot each other through springs, as proposed by Burridge and Knopoff in 1967 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Burridge (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The aseismic zone at depth is represented by a block whose basal friction responds to a slip-hardening law, and the seismogenic zone is represented by a block whose basal friction follows a slip-weakening law (figure 8a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Researchers then observed that for the seismogenic zone, the slip accumulates in ‘steps’ (figure 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This is expressed by jagged variations in the shear stress, which is accumulated over long periods of time and then released in a few seconds (figure 8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 11 seismogenic zone creep Slip-weakening friction Slip-hardening friction frictional surface F (constant pulling force) A Time Position of the slider = cumulative slip point A Slip-weakening Model Inside the seismogenic zone Slip-hardening Model Inside the creeping zone stress accumulation stress drop Slip-weakening Model Slip-hardening Model Time Shear stress a) c) d) b) Lorem ipsum τd s Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Modeling of a subduction using the block-spring method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' a) chematic representation of a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' b) Conceptual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' c) Accumulation of slip over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' c) State of shear stress over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' We then speak of a stress-drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' For the aseismic zone, after going through a plateau, which corresponds to the time required for the shear stress to reach the block’s value of resistance to slip, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', τ s d (figure 8d), the slip accumulates continuously and therefore there is indeed creep (figure 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 Modeling the seismic cycle align=left, leftmargin=2em, itemindent=0pt, labelsep=0pt, labelwidth=2em ] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1 Shifting to the rate-and-state law As discussed in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3, while the earlier model makes it possible to reproduce the esential steps that lead to the seismic slip, it does not allow multiple events to be chained, since µ does not return to its static value µd (figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' On the other hand, the R&S law, with the state variable θ, takes into account the healing of the fault plane (figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' If we go back to the spring-block slider model and replace the slip weakening friction law with a rate-and-state friction law, it is possible to derive a new instability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In this second case, during the acceleration phase (2a in figure 9), the slope of Ffric is approximately equal to σeff(b − a)/Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Consequently, for an instability, and potentially an earthquake, to be generated, we must have the following relation: k < kc ≈ ���� σeff(b − a) Dc ���� (16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 Implications for the nucleation size of earthquakes To move from the spring- block slider model to a slightly more realistic Earth model with elastic behavior, we use 12 slope � σeff (a-b)/ Dc Fspr (+) / Ffric (-) 1 stick 2a slip - acceleration 2b slip-deceleration Fspr exceeds Ffric here, so block accelerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Now there is an initeral force too!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Fsprand Ffricare equal here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' but there is still the initial force!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Block velocity slows to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Spring starts to strech again, setting up the next earthquake 1 stick spring is streching Ffric initially decreases then stays constant forces are equals here Fspr which pulls to the right, is decreasing Dc Displacement of block Block not moving Block not moving δ slope =-k Fault is “healing” and Ffric increases as slip slows and asperity contacts age EARTHQUAKE INTERSEISMIC INTERSEISMIC Ffric is larger than Fspr here, so block slows down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Fsprand Ffricare equal here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' but there is still the initial force!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Assessment of forces for the block-spring model with a rate-weakening friction law (Rate- and-state law) elasticity to determine the k value of an elliptical crack: k = G (1 − ν)L (17) where G is the shear modulus, ν is the Poisson’s ratio and L is the length of the zone that slips over the fault plane (figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In this case, the instability occurs when the decrease in the frictional force is greater than the decrease in elastic force, and equation 16 is rewritten as: G (1 − ν)L < kc ≈ ���� σeff(b − a) Dc ���� (18) Consequently, the zone that slips must be greater than a critical size Lc in order to become unstable and generate earthquake nucleation: L > Lc ≈ ���� DcG (1 − ν)σeff(b − a) ���� (19) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3 Continuum model In his seminal 1993 article Rice (1993), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Rice highlights the importance of moving from ”spring-block slider” models to continuous medium models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' He demonstrated, notably, that “While the equations of Newtonian dynamics are solved exactly in these Burridge-Knopoff models, it has not been generally acknowledged that the dynamical solution for rupture along a chain of lumped masses,or a string of concentrated mass in the continuous limit, bears a presently uncertain relation to dynamical solutions for rupture along a fault embedded in a surrounding elastic continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' For example, the 13 dx1 dx2 fault plane slipping elliptical patch of fault length L elastic rock with propeties G and ν Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Nucleation model response of B-K models to an instantaneous change in stress τ along the rupture is an instantaneous change in the acceleration ∂2δ/∂t2, but there is no instantaneous change in ∂δ/∂t.” This is true, on the other hand, in continuum models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The other major drawback is “Also, since there is no analogue to energy radiation as seismic waves in the normal implementation of the B-K models (an exception is the recent work of Knopoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' [1992]), all potential energy lost to the system during a rupture is fully account- able as frictional work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' the same is not true for rupture in a continuum.” It is therefore essential to highlight, in this text, that while the block-spring model makes it possible to qualitatively reproduce the phenomena observed in nature, it is essential to shift to a continuum model if we wish to develop robust numerical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Interested readers can consult The mechanics of faulting: from laboratory to real earthquakes Bizzarri & Bhat (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 3 A more complex physical reality align=left, leftmargin=2em, itemindent=0pt, labelsep=0pt, labelwidth=2em ] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1 Spatial and temporal variability in the slip mode on faults Until recently the deformation in fault zones, in the brittle part of the crust, was attributed either to earthquakes or to the slow, continuous slip during the inter-seismic period (creep) or post-seismic period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This latter phenomenon is called the afterslip and corresponds to a logarithmic acceleration in the aseismic slip on the fault, which can be observed after large earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' However, this paradigm of two ’extreme’ behaviors is being questioned today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Advances in technology and methodology in the field of geodesy and in seismology have significantly improved our capacity to measure deformation rates and given us higher res- olutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' These observations have enabled us to document a large variability in the slip dynamic in the seismogenic zone (figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Faults may have chiefly seismic behavior, have 14 + + + + + + + + + + + + + + + + + + + ++ creep ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' transition zone creep seismogenic zone upper plate Seismic Signals Geodetic Signals 400 nm/s 5 seconds Low Frequency Earthquake Few seconds, M<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 e) b) locked patch, slip from kinematic inversion Megathrust Earthquake several 10s of m in few seconds 50 cm 1 Year creepmeter Creep, upper segment ~ few cm in 1 year 1 Year a) 30 cm 600 nm/s 10 seconds Megathrust Earthquake Few seconds to mins, M<9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 d) 10 Days c) Deep Slow Slip Svent ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 mm in 1 day 5 mm 50 nm/s 200 seconds Very Low Frequency earthquake Tens of seconds, M<4 g) 600 nm/s 200 seconds Non Volcanic Tremors Seconds to hours, M<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 f) stable creep Generated by microcraks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' swarm of low frequency earthquakes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Représentation schématique d’une zone de subduction et de la distribution des sources potentiellement responsables des phéno- mènes observés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' GEODESIE : a) fluage enregistré par un extensomètre, b) glissement durant un séisme majeur , c) slow slip events (SSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' SISMOLOGIE : sismogrammes enregistrant (d) séismes, (e) Low Frequency Earthquakes, (f) non volcanic tremors, et (g) Very Low Frequency Earthquakes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Les patchs rouges représentent les aspérités “bloquées" pendant la période intersismique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Les isocontours mauves représentent le glissement cosismique qui aurait rompu plusieurs patchs adjacents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Les données en a) et b): [THO 14] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' c), d), e), f) : [PEN 10] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' wFreguency rthguake(LFE) EarthquakeVeryLow Frequency Event (VLF)Tremor15 a slow, stable slip Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2014a) or a transient slip Rousset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In ad- dition to this, one of the most significant discoveries in the last decade has been revealing the existence of ’slow earthquakes’ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Chapter 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' These encompass several phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Slow slip events rupture the fault very slowly over several hours or even days, at velocities that are higher than the inter-seismic creep (cm/yr), but slower than earthquakes, such that no detectable seismic waves are radiated Dragert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' They are generally (though not always) accompanied by weak seismic signals of a long duration (a few minutes to a few weeks) called non volcanic tremors Obara (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Low frequency earthquakes, with a duration close to a second, and Very low frequency earthquakes, which can last a hundred seconds, are commonly observed within non volcanic tremors Ide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' As a result, it is known today that slip velocities on faults cover a continuum going from a millimeter per year to a meter per second Peng & Gomberg (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This is therefore an essential parameter to take into consideration when modeling active faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' However, the physics behind the processes that govern this behavior is still unknown and is the subject of much active debate in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In addition to the large range of deformation velocities there is a spatial and temporal variability in the slip mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Contrary to what the schematic representation of figure 11 might suggest, the phenomena described here are not restricted to a specific depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' On some faults creep may be recorded over the entire seismogenic zone i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' from the surface up to the maximum depth where earthquakes are observed Titus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Further, while slow earthquakes were first located beyond the seismogenic zone Obara (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Ide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2007), non volcanic tremors and slow slip events have recently been observed at epths of less than 10 km, as well as in the sub-surface Ito & Obara (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Outerbridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Moreoever, geodetic data has shown that the seismic or aseismic behavior is not necessarily stable over time, and that the same zone may creep and slide seismically Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' These observations lead to two hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (1) These different phenomena can occur under varied pressure/temperature conditions and/or result from various deformation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2) They correspond to particular mechanical and rheological properties, but which vary over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Consequently, they also vary over space, depending on what seismic cycle phase the observed site is undergoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 Additional mechanisms that can come into play during earthquakes The standard formulation of the rate-and-state law, (section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5), allows a numerical re- production of a large number of the phenomena discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' However, this formulation was based on slip velocity experiments ranging from 10−9 to 10−3 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' While comparable to aseismic velocities (10−10 to 10−9 m/s), they are still slow when compared to seismic velocities (∼ 1 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' There is increasing experimental and theoretical proof that larger slip velocities and quantity of slip also come into play Lapusta & Barbot (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This has the ef- fect of drastically reducing the dynamic friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Wibberley and co-authors Wibberley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2008) have compiled laboratory values for different kinds of rocks and at different loading velocities (figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The lack of experimental data on the properties of friction that are applicable to earth- quakes is due to the difficulty of carrying out experiments in conditions similar to earth- quakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A laboratory experiment that would reproduce the conditions that exist during 16 Typical slow-slip values Earthquakes Plate motion mm/yr cm/yr m/s Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Dependence of the coefficient of dynamic friction, in a continuous regime, on the slip velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Figure modified as per Wibberley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' seismic slip would simultaneously involve high slip rates (1-10 m/s), with large displace- ments (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='1-20 m), a resulting effective normal stress (50-200 MPa), high pore pressure (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='4 to 1 times the normal stress) and high temperature (ambient temperatures of 100 to 300◦C, but potentially as high as 1500◦ C in the slip zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Although considerable progress has been made over the last decade, there is as yet no device that is capable of simultaneously responding to all these requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' It is therefore necessary to compromise on one or more factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Tullis and Schubert highlighted this difficulty and proposed a complete review of the processes that could lead to substantial reductions in the friction coefficient with respect to its typical experimental value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='6 Tullis & Schubert (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The proposed mechanisms include: dynamic reduction in the normal stress or loss of contact due to the vibrations perpen- dicular to the interface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' dynamic reduction in the normal stress due to the contrast in elastic properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' or permeability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' on either side of the fault,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' acoustic fluidization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' elasto-hydrodynamic lubrication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' thermal pressurization of pore fluids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' pressurization of pore fluids induced by the degradation of minerals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' local heating/melting of the point of contact between the asperities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' lubrication of the fault through fusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' in response to frictional processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' lubrication of the fault through the creation of a thixotropic silica gel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' superplastic deformation of fine grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' These highlight the difficulty of proving which mechanism is responsible for the observed ex- perimental behavior and to design experiments that can clearly prove or refute a mechanism proposed in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Nonetheless, since it is likely that one or more of these processes is acti- vated at high slip rates, the rate-and-state law described in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 does not adequately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='8 田 田 田 M V 田 芒 V 田 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='6 0 Friction coefficie G AV Peridotite 20 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2006b] Gabbro 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa [Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2008] G Tonalite 20 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2006b] 0 Clay-rich gouge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='6 MPa [Mizoguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2007] Granite 2 MPa [Dieterich 1978] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='4 A 田 Aplite 5 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' unpublished] Westerly granite 5 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2004] Feldspar 5 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' unpublished] Westerly granite 5 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' unpublished] G Feldspar1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' unpublished] 会 Serpentinite 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa [Hirose and Bystricky 2007] Carrara marble 12-15 MPa [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' in prep] V Serpentinite 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa [Hirose and Bystricky 2007] Carrara marble <10 MPa [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' in prep] Serpentinite 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa [Hirose and Bystricky 2007] Carrara marble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='25 MPa [Di Toro unpublished] V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 Serpentinite 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa [Hirose and Bystricky 2007] Calcite gouge [Shimamoto and Logan 1981] Serpentinite 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='5 MPa [Hirose and Bystricky 2007] Calcite gouge 100 MPa [Morrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2000] Serpentinite 25 MPa bare surface [Reinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1992] Dolomite marble 75 MPa[Weeks and Tullis 1985] 0 Serpentinite 25 MPa gouge-present[Reinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 1992] Dolomite marble 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 MPa [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' in prep] Quartz sandstone 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='7 MPa [Dieterich 1978] 口 Dolomite marble 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2 MPa [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' in prep] Novaculite 5 MPa [Di Toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 2004] Dolomite gouge [Shimamoto and Logan 1981] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0 1 6 1 1 11 10 6- 8 7 5 4 3 2 1 0 Log slip-rate (m s-1)17 reproduce this strong fall in the coefficient of dynamic friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Indeed, for seismic velocities (∼ 1 m/s) is a typical value for (a − b) equal to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='005, we obtain a µd of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Further, based on laboratory experiments, the effectve µd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=', τ/σeff, can reach very low values (0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='2) during co-seismic slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This observation has many implications for our understanding of the mechanism of earthquakes: on the amplitude of the stress drop, on the propensity of earthquakes to propagate in pulse form, on the amplitude of ground movements, and on the orientation of stresses in the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Lapusta and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Barbot propose two ways of modifying the rate-and-state law to take into account these additional weakening mechanisms Lapusta & Barbot (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Interested readers may refer to their publication for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='3 Going beyond the elastic Earth model Many ground studied, geophysical observations, and laboratory experiments have high- lighted the strong coupling that exists between the main rupture plane and the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The faults zones are not made up only of a major plane where the majority of slip occurs, but also make up a complex group, surrounded by a zone where surrounding rock is fractured intensively (figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Seismic ruptures result in damage around the faults with an exponential decrease in the density of microfractures perpendicular to the main slip plane Anders & Wiltschko (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Mitchell & Faulkner (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' The damage modifies the microstructure and changes the elastic properties of the rocks at the level of the fault breccia and in the adjacent medium Walsh (1965a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Faulkner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' These changes, in return, modify the extension an dynamic of the rupture as well as the radiation of seismic waves Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' They also influence seismic processes during the post-seismic period, such as aftershocks, with the minimum size of the nucleation zone depending chiefly on the elastic modulus Rubin & Ampuero (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In their experimental study, Gratier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2014) have also demonstrated that the co-seismic damage would promote aseismic slip through pressure-dissolution, thus explaining the afterslip recorded after large earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Co-seismic damage also increases permeability (figure 13e), which results in a variation in the fluid pressure Sibson (1994) that modifies the fault’s resistance to slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Geophysical observations suggest that this effect is transient (figure 13d), because a gradual and partial recovery of the elastic properties after the earthquake has been recorded Hiramatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Froment et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' This evolution is probably related to the healing of microfrac- tures and faults through the precipitation of dissolved substances, products of alteration and/or the development of clayey minerals Mitchell & Faulkner (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In their model, den Hartog & Spiers (2014) propose that the compaction through pressure-dissolution leads in turn to the recovery of seismogenic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Moreover, several studies have demonstrated the influence of the properties of the sur- rounding rock on the behavior of faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Audet and co-authors have shown a direct rela- tionship between the physical properties of the interlocking plate in the subduction zone and the recurrence of slow earthquakes Audet & Burgmann (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In my microstructure study of Taiwan’s longitudinal valley fault, Thomas and co-authors were able to demonstrate the aseismic behavior of the fault was controlled by inherited microstructure Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2016) looked at the influence of the ’maturity’ of the faults on the accumulation of slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' A study of 27 earthquakes concluded that the more damage the fault presents (mature fault), the greater the quantity of slip during an earthquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 18 Fluid flow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Pore Pressure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Alteration ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Elastic properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Temperature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Healing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Permeability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' creep ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' transition zone creep locked zone 0 50 100 150 200 2 5 10 15 20 Distance from the fault core (m) Log cracks Density (#/mm) Caleta Coloso Fault Blanca fault Background microfracture Damage Density a) at peak stress of cycle at 60 % failure stress 10-17 10-18 10-19 Permeability (m ) 2 Permeability changes with damage density Microfracture density (#/mm) 2 4 6 10 12 14 16 8 e) 70 66 16 8 0 58 54 62 Number of cycles Poisson’s ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='30 Microfracture density (#/mm) Young’s modulus (GPa) 2 4 6 1 3 7 9 8 5 Elastic Properties changes with damage density 40 MPa 60 MPa 80 MPa 100 MPa c) Temperature Variation of (a-b) Temperature (°C) 0 100 200 300 400 500 Illite gouge (a-b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='03 b) Damage Zone Alteration Fault core Fault breccia 250 300 350 Julian day in 2008 10 8 6 4 2 0 dV/V (%) Healing Temporal changes in seismic velocity d) SEISMIC ASEISMIC Westerly granite Gofar transform fault Westerly granite Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Représentation schématique, du point de vue de la mécanique, d’une zone de subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' L’interface entre la plaque plongeante et la plaque chevauchante est une zone complexe qui comprend une brèche de faille, des plans de glissement principaux, et une zone d’endommagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' La deforma- tion dépend de plusieurs paramètres qui ont leur propres évolutions temporelles durant le cycle sismique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (a) L’endommagement décroit de façon exponen- tielle avec la distance à la faille [MIT 09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (b) Les paramètres de friction (a − b) dépendent de la température [HAR 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (c) Les propriétés élastiques varient avec l’endommagement [FAU 06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (d) La vitesse des ondes sismiques changent après un séisme [FRO 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Le trait pointillé marque la date d’un séisme de Mw 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (e) La perméabilité évolue en fonction de la densité de microfracture [MIT 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' 19 4 Transition towards a new generation of models The usual way of looking at the fault restricts the deformation in the brittle part of the crust to slip along the interface (fault plane), loaded with creep at depth, whose behavior is controlled by its frictional properties Scholz (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' According to these properties, when the resistance threshold is exceeded, the stress accumulated when the fault is locked is released through seismic slip or creep, or again during slow earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Further, as the previously cited studies have highlighted, while the behavior of the fault zones is intrinsically related to the properties of the main slip plane, it also depends on the properties of the surrounding rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' In parallel, the displacement on the faults induces a modification of the physical properties of the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' these observations suggest the existence of a second ‘cycle’ where the properties of the fault zone evolve with respect to the slip dynamic, which in turn influences the deformation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' However, the majority of models used today do not take this complex feedback into ac- count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' By attributing constant properties (pressure, temperature, petrology, microstructure) that do not evolve with deformation, we neglect to take into account how seismic/aseismic fault behavior is impacted by temporal variations of the physical properties of the volume and the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' It is thus useful to develop a new generation of models that take into account spatial-temporal evolution of physical properties in fault zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' New models are being developed and have already shown the importance of these interactions from a seismic point of view Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' (2017b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Thomas & Bhat (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfo_1S/content/2301.01605v1.pdf'} +page_content=' Okubo et al.' metadata={'source': 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Schilberth,1, 2 M.-C. Jiang,3, 4 S. Minami,5, 6 M. A. Kassem,7 F. Mayr,1 J. Deisenhofer,1 T. Koretsune,8 +Y. Tabata,7 T. Waki,7 H. Nakamura,7 G.-Y. Guo,3, 9 R. Arita,4, 10 I. Kézsmárki,1 and S. Bordács2 +1Experimentalphysik V, Center for Electronic Correlations and Magnetism, +Institute for Physics, Augsburg University, D-86135 Augsburg, Germany +2Department of Physics, Institute of Physics, Budapest University of Technology and Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary +3Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan +4RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako 351-0198, Japan +5Department of Mechanical Engineering and Science, +Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan +6Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan +7Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan +8Department of Physics, Tohoku University, Sendai 980-8578, Japan +9Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan +10Research Center for Advanced Science and Technology, +University of Tokyo, 4-6-1 Meguro-ku, Tokyo, 153-8904, Japan +(Dated: February 1, 2023) +Giant anomalous Hall effect (AHE) and magneto-optical activity can emerge in magnets with topologically +non-trivial degeneracies. However, identifying the specific band structure features like Weyl points, nodal lines or +planes which generate the anomalous response is a challenging issue. Since the low-energy interband transitions +can govern the static AHE, we addressed this question in the prototypical magnetic Weyl semimetal Co3Sn2S2 +also hosting nodal lines by broadband polarized reflectivity and magneto-optical Kerr effect spectroscopy with a +focus on the far-infrared range. In the linear dichroism spectrum we observe a strong resonance at 40 meV, which +also shows up in the optical Hall conductivity spectrum and primarily determines the static AHE, thus, confirms +its intrinsic origin. Our material-specific theory reproduces the experimental data remarkably well and shows that +strongly tilted nodal line segments around the Fermi energy generate the resonance. While the Weyl points only +give vanishing contributions, these segments of the nodal lines gapped by the spin-orbit coupling dominate the +low-energy optical response. +Topological Dirac and Weyl semimetals have received much +attention, since at low-energies their electrons mimic relativis- +tic particles [1]. Moreover, topological semimetals with higher +dimensional degenerate manifolds, such as nodal lines and +even planes, have also been predicted and observed, which host +quasiparticles that are unprecedented in particle physics [2–5]. +These peculiar band structure features give rise to e.g. excep- +tionally high mobility [6, 7], chiral anomaly [8, 9], Fermi arcs +and drumhead surface states [10–13], and unusual quantization +of orbital motion in a magnetic field [14, 15]. +Recently, the search for such topological band features in +magnetic materials has become a hot topic. In magnets, the +topological nodes can be controlled by magnetic fields [16–18], +they induce exotic domain wall states [19, 20] and generate +enhanced anomalous Hall effect (AHE) [1, 21]. The intrinsic +AHE being proportional to the Berry-curvature integrated over +the Brillouin zone (BZ) has particular importance as it is a +direct consequence of the non-trivial band topology [22]. At +the heart of these phenomena, there is the interplay between +the magnetic order and the band structure mediated by the +spin-orbit coupling (SOC). In the ordered state, the broken +spin-rotation symmetry may either reduce the degeneracy of +the manifolds, e.g. transform a nodal line into Weyl points +[23, 24], or completely gap out the nodes, which may stabilize +a topological insulator phase [25]. Therefore, from the many +band structure features, pinpointing those responsible for the +anomalous responses is highly desirable. +This is an especially important question in the prototypi- +cal magnetic Weyl semimetal Co3Sn2S2 with individual Weyl +points remaining degenerate from SOC gapped nodal loops. Its +crystal structure (space group R¯3m) consists of an ABC-type +stack of Co3Sn kagome layers (see inset in Fig. 1(a)), and +belongs to the family of shandites [26]. Below Tc = 177K, +a ferromagnetic order develops with the moments aligned to- +wards the c axis at low temperatures. Just below the transition, +an anomalous magnetic phase with non-collinear order was +proposed [27, 28], but more recent experiments suggest that +the domain configuration changes instead [29, 30]. Due to its +kagome structure, Co3Sn2S2 possesses nontrivial electronic +topology. In addition to flat bands [31, 32], non-relativistic den- +sity functional theory (DFT) calculations propose nodal loops +on high-symmetry planes of the BZ, which are gapped upon in- +cluding SOC, each leaving behind a pair of Weyl nodes [33, 34]. +Angle-resolved photoemission spectroscopy (ARPES) stud- +ies confirmed the existence of Fermi arcs in this system [35] +and chiral edge modes were found in scanning tunneling mi- +croscopy (STM) [36]. The Berry curvature accumulated by the +anticrossing line is claimed to be the source of large anoma- +lous Hall and Nernst effects in this material [23, 33, 34]. The +magnitude of the former reaches as high as 1200Ω−1cm−1 and +is therefore comparable to the AHE in the related compound +Fe3Sn2 [37, 38], in which, as demonstrated recently, only a +fraction of the intrinsic AHE can be attributed to twisted nodal +lines [39]. In addition, although the DFT band structure of +arXiv:2301.13726v1 [cond-mat.str-el] 31 Jan 2023 + +2 +Co3Sn2S2 is relatively simple close to the Fermi level, it is to +date unclear which band structure feature dominates the AHE: +the gapped nodal loop or the Weyl points. +Here, we address this fundamental question and determine +the full optical conductivity tensor of Co3Sn2S2 by polarized +infrared reflectivity and magneto-optical Kerr effect (MOKE) +spectroscopy (For simplicity and readability, we omit the ex- +plicit frequency dependence for the rest of the paper): +ˆσ(ω) = +� +� +σxx(ω) +σxy(ω) +0 +−σxy(ω) σxx(ω) +0 +0 +0 +σzz(ω) +� +�. +(1) +Our results indicate that linear dichroism, namely the ratio of +the conductivity in the kagome plane, σxx and out-of-plane, σzz, +is a sensitive probe of topological features of layered materials. +We find a resonant enhancement of the linear dichroism due +to transitions along the gapped nodal line. We reveal a giant +magneto-optical optical activity in the same energy range by +extending MOKE spectroscopy down to ¯hω = 25 meV. Specif- +ically, we observe 1) a resonance peak at 40 meV in the Hall +conductivity spectrum, σxy, which has not been detected be- +fore, and 2) directly capture the fingerprints of the nodal loop +in the optical conductivity without any extrapolation of σxy. +Complemented by ab-initio calculations, we analyse the mo- +mentum space distribution of the Hall spectral weight, which +allows us to disentangle the contributions of the gapped nodal +line and Weyl points. +Polarized reflectivity spectra were measured on polished ab +and ac surfaces of single crystals with a lateral size of ∼5 mm +and ∼3 mm, respectively over the range of 0.01 – 3 eV. The di- +agonal optical conductivity spectra were obtained by Kramers- +Kronig-transformation of the reflectivity. The magneto-optical +Kerr rotation, θ and ellipticity, η were measured at near nor- +mal incidence on the same ab-cut crystal in ±0.3 T and were +antisymmetrized with respect to the field. The Hall conduc- +tivity spectra were calculated using the complex Kerr rotation +(0.025 – 3 eV) according to +θ +iη = − +σxy +σxx +� +1+i 1 +ε0ω σxx +, +(2) +where ω is the angular frequency of the photon and ε0 is the +vacuum permittivity [40]. +The reflectivity and MOKE spectra measured between 10- +200 K are included in Fig. S1. The out-of-plane reflectivity +remarkably differs from the in-plane spectrum, the latter being +in agreement with earlier reports [32, 41]. In the overlapping +energy range, the MOKE spectra agree with those published in +Ref. 42. Importantly, our Kerr rotation and ellipticity obey the +Kramers-Kronig relation and fulfil the magneto-optical sum +rule, requiring that both parameters approach 0 for ω → 0. +We show the low-energy spectrum for each independent +component of the optical conductivity tensor in Fig. 1 (for a +broad energy range see Fig. S2). The corresponding static +conductivity values are shown for comparison and agree well +with the respective spectra at the low-energy cutoff. In panel +Co +Sn +S +0 +20 +40 +60 +Re +xx (102 +1 cm +1) +(a) +0 +20 +40 +60 +Re +zz (102 +1 cm +1) +(b) +0 +5 +10 +15 +20 +Im +xy (102 +1 cm +1) +(c) +0.0 +0.05 +0.1 +0.15 +0.2 +Energy (eV) +0 +5 +10 +15 +20 +Re +xy (102 +1 cm +1) +(d) +Theory +10 K +20 K +40 K +60 K +80 K +100 K +120 K +140 K +160 K +170 K +172 K +174 K +176 K +178 K +180 K +200 K +FIG. 1. Comparison of the experimental conductivity spectra mea- +sured between 10 and 200 K (colored lines) and the theoretical DFT +spectra (black lines) calculated as described in the text. a), b), c) and +d) respectively show the real parts of the diagonal, Re σxx & Re σzz, +as well as, the imaginary and real part of the off-diagonal conductivity +spectra, Im σxy and Re σxy. For comparison, the static conductivity +values are plotted as colored squares at zero energy. +(a), the real part of σxx exhibits a Drude-like increase towards +zero energy responsible for the static conductivity. At 30 meV, +a peak is forming below 100 K, separated well from the free +carrier response. For even higher energies, we find a small +temperature dependent hump around 0.25 eV and a step edge +around 0.6 eV before the conductivity becomes flat without +significant temperature dependence. These features agree with +earlier reports [32, 41, 42]. +The out-of-plane conductivity spectrum, σzz in panel (b), +strongly deviates from σxx. Most strikingly, no sign of a Drude +peak is observed down to our low-enegy cutoff, and the dc + +3 +conductivity is also much lower for this direction, indicated by +the colored points at zero energy. Therefore, we suspect that +within the kagome planes, the strong orbital overlap between +Co-sites can produce a coherent conduction, manifested in the +Drude term. In contrast, the transport is likely due to incoherent +hopping perpendicular to the planes [43]. At 200 K, we find a +peak at 40 meV, which shifts to smaller energies upon lowering +the temperature until it eventually splits in two below 100 K. At +higher energies, we find a minimum at 0.4 eV, which becomes +sharper at low temperatures, and a step edge at 0.6 eV. For even +higher energies, σzz is featureless though slightly increasing +without distinct temperature dependence, similar to σxx. +In Fig. 1(c) & (d), the Hall conductivity spectrum σxy shows +a strong resonance at 40 meV, in coincidence with the in- and +out-of-plane diagonal components and their ratio. We empha- +size that this far-infrared range has not been covered so far, +while the higher energy part of the spectra agree very well +with Ref. 42. Both its imaginary and real parts exhibit a large +enhancement towards low temperatures, where the peak in the +real part overshoots the dc-AHE below 60 K with a magni- +tude as high as 2000 Ω−1cm−1 at 10 K. The good agreement +between the low-energy tail of the real part of σxy and the dc- +AHE together with the formerly published featureless THz data +[42], suggests that there are no further excitations in the narrow +uncovered energy interval. Furthermore, as the scattering rate +obtained from the Drude peak for σxx is below the cutoff for +σxy, we conclude that the giant anomalous Hall conductivity of +Co3Sn2S2 has dominantly intrinsic origin and it is generated +by the interband resonance observed here for the first time. +In order to reveal the microscopic origin of the observed +spectral features, we performed ab–initio calculations provid- +ing all symmetry allowed elements of the conductivity tensor +[44]. The theoretical spectra are coplotted with the experiment +in Figs. 1 and S2. Disregarding the intraband contribution +which is not included in the theory, the spectra qualitatively +reproduce all conductivity components on a large energy scale, +although the spectral features appear slightly shifted to higher +energies compared to experiment, which may indicate correla- +tion effects [32]. +We quantified the linear dichroism for in- and out-of-plane +polarizations by calculating Re σzz/Re σxx from the spectra +shown in Fig. 1. As demonstrated by Fig. 2, the theory agrees +with the experiment remarkably well, especially at higher tem- +peratures. We find that around the resonance at 40 meV, σzz +becomes three times larger than σxx. As the temperature is +decreased in the experiment, some of the spectral weight splits +and moves to lower energies. Although this is not properly +captured by the DFT calculations the overall tendency, that σzz +is stronger at the resonances, remains valid. +The Hall conductivity is also well reproduced by the the- +ory as shown in Figs. 1 and S2(c) & (d). The imaginary part +shows a sudden increase at 40 meV, where the resonance is +observed in the experiment, though it is not that pronounced. +Importantly, the theory properly captures the 35.9 meV peak in +Reσxy. The experimental feature is somewhat sharper like in +the case of the imaginary part, which may again be related to +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Energy (eV) +0 +1 +2 +3 +4 +Re +zz/Re +xx +0 +1 +2 +3 +0 +2 +4 +FIG. 2. Optical anisotropy spectra Re σzz/Re σxx with the same +colorcode as in Fig. 1. The inset shows the spectra over a broader +energy range. +electronic correlations [32]. The dc–extrapolation of the theory +yields the same AHE as the magnetotransport measurements. +Backed with this remarkable agreement, we now analyse the +band structure origin of this giant optical Hall effect signal. +We deduced the momentum decomposition of the off- +diagonal conductivity by introducing the Hall spectral weight +Hxy(ω,k) as σxy(ω) = ie2/¯hV +� +BZ Hxy(ω,k). (For details see +Eq. S1 and text of SM). Fig. 3(b) shows the Hall spectral weight +of the 35.9 meV low-energy peak in Re σxy on one of the mir- +ror planes containing the nodal line and the Weyl points as +indicated by the rainbow line and black labels, respectively +[34, 45]. The black line is the border of the BZ, the green lines +plot the Fermi surface, and the colormap presents the spectral +weight. For better orientation, panel (a) plots the nodal lines +where the coloring encodes the position of the band crossing +relative to the Fermi energy. +We find several hot spots of Hall conductivity on the slice. +Interestingly, there is no significant contribution from the Weyl +points as they are located 60 meV above the Fermi energy and +cannot contribute to the response in the range below [34, 45]. +Instead, we find regions close to Γ and in the vicinity of A with +positive and negative weights right next to each other. Since +the conductivity is determined by integration over the BZ, the +opposite weight from these points cancel to a large extent. This +leaves the large positive patch on the Γ − A line giving the +dominant contribution. Aside from the regions around Γ, we +always observe a hot spot when the nodal line comes close to +or crosses the Fermi energy (light green segments of the loop), +hence the hotspots always connect two close regions of the +Fermi surface (dark green lines). +Although the hot spots seem to have a similar origin, they +behave differently in producing optical weight. In order to see +the underlying shape of the bands, we introduce two non-high- +symmetry points A and B, to plot the band structure along cuts +through the hot spots, shown in Fig. 3(c) with and without SOC. +The grey shading indicates the relevant low-energy range and +is shaded in the same fashion in Fig. 1. Along the B−A line, +we cut through a hot spot with subsequent positive and negative + +4 +L +A +B +Γ +L +A +(c) +Energy (eV) +0.1 +0.2 +-0.1 +-0.2 +0.0 +SOC +Spin Up +Spin Dn +140 +70 +0 +-70 +-140 +Energy (meV) +(a) +kx +kz ky +T +U +L +Γ +kz +Γ +T +U +L +A +B +Weyl +Weyl +Hxy(ω,k) (102 Å2) +0 +5 +10 +-10 +-5 +ky +(b) +FIG. 3. (a) Location of the nodal lines in the Brillouin zone. The colorscale encodes the position relative to the Fermi energy. (b) The Hall +spectral weight of the calculated 35.9 meV peak on the mirror plane containing the nodal line and Weyl points. (c) Bandstructure along the main +contributing areas of the real part of the off-diagonal optical conductivity. The grey shading highlights the energy range below 50 meV where we +expect contributions to the peak in Re σxy. The non-high-symmetry points A and B are (0.0, 0.4002, 0.301) and (0.0, 0.7373, 0.0) in reciprocal +lattice units. +weights. Here, the two red spin up bands forming the nodal line +are close to the Fermi energy, so with SOC, one of the bands is +filled and the other is empty in between the crossings, allowing +the optical transitions. Importantly, the tilt of the crossing +points is opposite, which is the reason for the different signs of +their contribution to σxy [46, 47]. Around Γ, we have a similar +situation where two red spin up band inversions appear with +opposite tilt. The SOC opens a gap, which allows transitions +once the upper band is above the Fermi energy, also producing +an optical Hall response. +The situation must be different along the A−Γ line, where +the large positive hot spot does not have a negative partner +nearby. Here, we observe only one strongly tilted crossing. +Again, with SOC the upper band is pushed above the Fermi +level enabling the transition. Due to the strong tilt, the two +bands stay nearly parallel over a relatively broad k-interval, so +the Berry curvature of the gapped nodal line is summed up +over a narrow energy range which produces the large patch +of optical weight. By inversion and 3–fold rotation symmetry, +we expect a total of 6 such spots in the BZ dominating the +anomalous Hall conductivity. +Since the interband transition from this feature also gives +rise to diagonal conductivity, we directly compare σxx and σxy +by calculating the Hall angle, ΘH = arctanRe(σxy/σxx), which +is shown in Fig. S4(b). At 40 meV, the giant off–diagonal +conductivity has almost the same magnitude as the diagonal +conductivity producing a very large Hall angle of 42.7◦. When +the Kubo formula is written using the circular momentum +operators +Im σxy = +e2π +4m2V ¯h ∑ +k,n,n′ +|f(εn(k))− f(ε′ +n(k))| +ωnn′ +δ(ω −ωnn′) +× +���⟨n,k|p+|n′,k⟩ +��2 − +��⟨n,k|p−|n′,k⟩ +��2� +, +(3) +σxy depends on the difference of the two circular components, +whereas σxx is given by their sum [48]. As a consequence, +the largest possible Hall angle for an interband transition is +45◦. This is realized when one matrix element is zero, which +indicates an almost fully polarised transition in the present +case, yielding a nodal line resonance. +In summary, we provide a showcase for disentangling the +contributions of various topological features to the AHE, and +identify the gapped nodal line as the main source of the giant +AHE in Co3Sn2S2. Facilitated by far infrared MOKE spec- +troscopy, we observe a low-energy magneto-optical resonance +for the first time. As the zero energy extrapolation of this +interband transition explains the static Hall conductivity, we +confirmed that the AHE is dominantly intrinsic. Our ab-initio +calculations are able to reproduce the experimental spectra +with remarkable accuracy, allowing a momentum and band de- +composition of the optical Hall conductivity. We find that the +Weyl points located 60 meV above the Fermi energy only yield +singular contributions in a small k-volume. By contrast, the +nodal line segments approaching or crossing the Fermi energy +produce large AHE hotspots after being gapped by SOC. In ad- +dition, we verify that the tilt of the nodal line is a crucial factor, +which can lead to pairwise cancellation or in contrast, produce +a nodal line resonance. Remarkably, the linear dichroism is +significantly enhanced by the nodal line resonance, leading to +a potentially new signature of topological states. Finally, we +note that this magneto-optical analysis is applicable for any +material where large AHE is suspected from topological bands. +Since in magnetic materials, the electronic topology may cou- +ple to the magnetic order, it is also a suitable tool to monitor +the effects of e.g. external fields for manipulating topological +properties. +The authors are grateful to Christine Kuntscher, Liviu Chion- +cel, and Artem Pronin for fruitful discussions. This work was +supported by the Hungarian National Research, Development + +0.14 + 1000 +0.07 +500 +0 +0 +-0.07 +←-500 +-0.14 +-1000 +kx10 +5 +0 +-5 +-10 +kx0.14 + 1000 +0.07 +500 +0 +0 +-0.07 +←-500 +-0.14 +-1000 +kx5 +and Innovation Office – NKFIH grants FK 135003 and Bolyai +00318/20/11 and by the Ministry of Innovation and Technol- +ogy and the National Research, Development and Innovation +Office within the Quantum Information National Laboratory +of Hungary. 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Marrazzo, +Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, +S. Poncé, T. Ponweiser, J. Qiao, F. Thöle, S. S. Tsirkin, +M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. +Mostofi, and J. R. Yates, Wannier90 as a community code: new +features and applications, Journal of Physics: Condensed Matter +32, 165902 (2020). +[61] M.-C. Jiang and G.-Y. Guo, Large magneto-optical effect and +magnetic anisotropy energy in two-dimensional metallic ferro- +magnet Fe3GeTe2, Phys. Rev. B 105, 014437 (2022). + +8 +SUPPLEMENTAL MATERIAL +Crystal growth for the magneto-optical measurements +A large single crystal of Co3Sn2S2 was grown in a cylindro- +conical shape (1 cm in diameter and 5 cm in length) from its +melt by a modified Bridgman method [49, 50]. About 10 g of +polycrystalline Co3Sn2S2 synthesized by a solid state reaction +was charged in a tipped glassy carbon crucible which was +sealed under vacuum in a quartz tube. The sealed ampule +was suspended by a Kanthal thread from the top to the hot +zone of a vertical tube furnace and heated over 30 hours up +to 1000◦C, kept for 6 h, and then slowly cooled over 72 h to +800◦C. After furnace cooling, a crystal has been removed from +the crucible and could be easily cleaved in the (001) plane. +Crushed parts were investigated by powder x-ray diffraction +(XRD), Laue x-ray spectroscopy and wave-length dispersive +x-ray spectroscopy (WDX), those indicated a single-phase +and high-quality grown crystal with stoichiometric chemical +composition of Co3Sn2S2. +Crystal growth for magnetotransport +Single crystals of Co3Sn2S2 were grown, in a flat hexagonal +shape, out of Sn flux as published elsewhere [51]. Lumps of +Co (99.9 % high purity chemicals) and grains of Sn (99.999 % +high purity chemicals) were mixed with S powder (99.999 % +nacalai tesque) in a molar ratio of Co : S: Sn = 8: 6: 86. A +mixture with a total mass of ≈ 15g was charged in Al2O3 +growth crucible. The growth crucible was covered by another +inverted Al2O3 crucible filled with quartz wool and both were +sealed under vacuum in a quartz tube. The ampule was fired in +a muffle furnace at 1050◦C for 6 hours and then slowly cooled +to 700◦C over 72 hours at which the ampule was removed from +the furnace. The flux was removed from the crystals via a rapid +decanting of the ampule followed by a subsequent spinning in +a centrifuge. The grown crystals were characterized by powder +XRD and WDX and the crystal orientations were determined +by Laue x-ray spectroscopy. +Reflectivity measurements +The reflectivity spectra of Co3Sn2S2 were obtained in a +Bruker IFS/66 FTIR-spectrometer for the MIR-VIS range and +a Bruker Vertex 80v for the FIR. The spectra were measured +in the frequency range 80-32000 cm−1 (0.01-4 eV) from room +temperature down to 10 K. As references, a silver and gold +mirror where used in the MIR-VIS and FIR experiments, re- +spectively. The optical conductivity σ1 was calculated by using +Kramers-Kronig analysis on the merged spectra. At this point, +the low-energy side of the reflectivity spectrum was extrapo- +lated by using a Hagen-Rubens law and the dc-conductivity, +while for the UV the reflectivity spectrum was extrapolated +with free electron behaviour setting in at 106 cm−1 and an +exponent for the interband regime of 1.5. +MOKE-spectroscopy +The broadband MOKE spectra were recorded in near-normal +incidence and were combined from several measurements in +different frequency ranges, employing grating and interfer- +ometer based spectrometers as described elsewhere [39, 52– +54]. Small permanent magnets provided a field of 0.3 T at +the sample position. Because of the large uniaxial anisotropy +in this material, the sample was warmed up over Tc between +measurements with reversed field direction to ensure proper +antisymmetrisation. +In the overlapping energy range, the MOKE spectra agree +with those published in Ref. 42. Beside the giant Kerr rotation +with a peak magnitude of -3.3 deg at 0.09 eV, we resolve a peak +around 50 meV in the ellipticity with a magnitude of 2 deg, +which was not detected before. +DFT calculations +The electronic structure of Co3Sn2S2 is calculated using +VASP code [55–57] based on the density functional theory. +The generalized gradient approximation of Perdew-Burke- +Ernzerhof was adopted for the exchange-correlation functional +[58]. The crystal structure for Co3Sn2S2 is of trigonal form +with a = 5.379 Å and α = 59.8658◦, which is the experimentally +determined lattice constant [59]. In the self-consistent band +structure calculations, Γ-centered k meshes of 24 × 24 × 24 +were used in the Brillouin zone integration. The optical prop- +erties are further evaluated using the Wannier functions, and +the Kubo-Greenwood formula [60]. +Below, we present the formula used for calculating Fig. 1(c) +and deducing the Hall spectral weight of a certain transition +energy plotted in Fig. 3(b). +σk,αβ(¯hω) = ie2 +¯hV ∑ +k,n,n′ +( fn′,k − fn,k)Re +� +εn′,k −εn,k +εn′,k −εn,k −(¯hω +iη) +� +Ann′,α(k)Ann′,β(k) += ie2 +¯hV ∑ +k +Hαβ(ω,k) +(S1) +The α and β are the indices in Cartesian coordinates, V is the +cell volume, and fn,k = f(εn,k) is the Fermi-Dirac distribution +function. ω is the optical frequency and η > 0 is the smearing +parameter. +Finally, Ann′,α = +� +un,k +��i∇kα +��un′,k +� +is the Berry connection +and Hαβ is the Hall spectral weight at certain transition en- +ergy. Importantly, if we apply ⟨v⟩ = ⟨p⟩/m and the relation +� +ψn,k +��v +��ψn′,k +� += −i/¯h · (εn′,k − εn,k)Ann′(k), we would arrive +at Eq. 3 of the main text. If we take ω and η as zero, we obtain +the ordinary Berry curvature for the anomalous Hall effect. + +9 +Energy (eV) +0.4 +0.6 +0.8 +1.0 +Reflectivityxx +(a) +Energy (eV) +0.4 +0.6 +0.8 +1.0 +Reflectivityzz +(b) +Energy (eV) +3 +2 +1 +0 +1 +Rotation (deg) +(c) +10 K +20 K +40 K +60 K +80 K +100 K +120 K +140 K +160 K +170 K +172 K +174 K +176 K +178 K +180 K +200 K +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Energy (eV) +2 +1 +0 +1 +2 +Ellipticity (deg) +(d) +FIG. S1. Spectra measured for several temperatures between 10 and +200 K of a) the reflectivity on the kagome plane and b) along the +stacking direction c) Kerr-rotation and d) ellipticity on the ab-plane +in the energy range up to 1 eV. +A fine mesh of 100 × 100 × 100 k points are applied during +the integration with good convergence. The Wannier functions +were constructed using Co d, Sn s, p, and S s, p orbitals with +a resulting tight-binding model well describing the DFT band +structures. Importantly, from the main text we notice that the +experimental low-frequency peak of the real part off-diagonal +optical conductivity Reσxy is nicely captured by the theoretical +calculations (Fig. 1(d)). + +10 +0 +20 +40 +60 +Re +xx (102 +1 cm +1) +(a) +0 +20 +40 +60 +Re +zz (102 +1 cm +1) +(b) +0 +5 +10 +15 +20 +Im +xy (102 +1 cm +1) +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Energy (eV) +5 +0 +5 +10 +15 +Re +xy (102 +1 cm +1) +(d) +10 K +200 K +Theory +FIG. S2. Conductivity spectra over a broad energy range. + +11 +(a) +(b) +FIG. S3. a) Calculated band structures along the high symmetry line with and without the spin-orbit coupling. The eigenvalues of the Wannier +tight-binding model are also plotted. b) Comparison of band structures over a large energy range calculated by density functional theory and +Wannier tight-binding model. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Energy (eV) +0 +10 +20 +30 +40 +H (deg) +Theory +10 K +20 K +40 K +60 K +80 K +100 K +120 K +140 K +160 K +170 K +172 K +174 K +176 K +178 K +180 K +200 K +FIG. S4. Hall angle spectra arctanRe(σxy/σxx) with a maximum of 42.7◦. + +6 +5 +DFT +- WANNIER +3 +Energy (eV +2 +A +B0.5 +0.4 +0.3 +0.2 +0.1 +Spin up +-0.1 +Spin Dn +SOC +-0.2 +annier +-0.3 +-0.4 +-0.5 +W +U \ No newline at end of file diff --git a/j9FST4oBgHgl3EQfHjj7/content/tmp_files/load_file.txt b/j9FST4oBgHgl3EQfHjj7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..abea0a83d9906057b79a4cbbf5264bee61a6a705 --- /dev/null +++ b/j9FST4oBgHgl3EQfHjj7/content/tmp_files/load_file.txt @@ -0,0 +1,1141 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf,len=1140 +page_content='Nodal line resonance generating the giant anomalous Hall effect of Co3Sn2S2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Schilberth,1, 2 M.' metadata={'source': 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+page_content=' Arita,4, 10 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Kézsmárki,1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Bordács2 1Experimentalphysik V, Center for Electronic Correlations and Magnetism, Institute for Physics, Augsburg University, D-86135 Augsburg, Germany 2Department of Physics, Institute of Physics, Budapest University of Technology and Economics, M˝uegyetem rkp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' H-1111 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Hungary 3Department of Physics and Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' National Taiwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Taipei 10617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Taiwan 4RIKEN Center for Emergent Matter Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 2-1 Hirosawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Wako 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Japan 5Department of Mechanical Engineering and Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Kyoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Nishikyo-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Kyoto 615-8540,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Japan 6Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Bunkyo-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Tokyo 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Japan 7Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Kyoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Kyoto 606-8501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Japan 8Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Sendai 980-8578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Japan 9Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' National Center for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Taipei 10617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Taiwan 10Research Center for Advanced Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 4-6-1 Meguro-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 153-8904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Japan (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 2023) Giant anomalous Hall effect (AHE) and magneto-optical activity can emerge in magnets with topologically non-trivial degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' However, identifying the specific band structure features like Weyl points, nodal lines or planes which generate the anomalous response is a challenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Since the low-energy interband transitions can govern the static AHE, we addressed this question in the prototypical magnetic Weyl semimetal Co3Sn2S2 also hosting nodal lines by broadband polarized reflectivity and magneto-optical Kerr effect spectroscopy with a focus on the far-infrared range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In the linear dichroism spectrum we observe a strong resonance at 40 meV, which also shows up in the optical Hall conductivity spectrum and primarily determines the static AHE, thus, confirms its intrinsic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Our material-specific theory reproduces the experimental data remarkably well and shows that strongly tilted nodal line segments around the Fermi energy generate the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' While the Weyl points only give vanishing contributions, these segments of the nodal lines gapped by the spin-orbit coupling dominate the low-energy optical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Topological Dirac and Weyl semimetals have received much attention, since at low-energies their electrons mimic relativis- tic particles [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Moreover, topological semimetals with higher dimensional degenerate manifolds, such as nodal lines and even planes, have also been predicted and observed, which host quasiparticles that are unprecedented in particle physics [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' These peculiar band structure features give rise to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' excep- tionally high mobility [6, 7], chiral anomaly [8, 9], Fermi arcs and drumhead surface states [10–13], and unusual quantization of orbital motion in a magnetic field [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Recently, the search for such topological band features in magnetic materials has become a hot topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In magnets, the topological nodes can be controlled by magnetic fields [16–18], they induce exotic domain wall states [19, 20] and generate enhanced anomalous Hall effect (AHE) [1, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The intrinsic AHE being proportional to the Berry-curvature integrated over the Brillouin zone (BZ) has particular importance as it is a direct consequence of the non-trivial band topology [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' At the heart of these phenomena, there is the interplay between the magnetic order and the band structure mediated by the spin-orbit coupling (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In the ordered state, the broken spin-rotation symmetry may either reduce the degeneracy of the manifolds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' transform a nodal line into Weyl points [23, 24], or completely gap out the nodes, which may stabilize a topological insulator phase [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Therefore, from the many band structure features, pinpointing those responsible for the anomalous responses is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' This is an especially important question in the prototypi- cal magnetic Weyl semimetal Co3Sn2S2 with individual Weyl points remaining degenerate from SOC gapped nodal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Its crystal structure (space group R¯3m) consists of an ABC-type stack of Co3Sn kagome layers (see inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1(a)), and belongs to the family of shandites [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Below Tc = 177K, a ferromagnetic order develops with the moments aligned to- wards the c axis at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Just below the transition, an anomalous magnetic phase with non-collinear order was proposed [27, 28], but more recent experiments suggest that the domain configuration changes instead [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Due to its kagome structure, Co3Sn2S2 possesses nontrivial electronic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In addition to flat bands [31, 32], non-relativistic den- sity functional theory (DFT) calculations propose nodal loops on high-symmetry planes of the BZ, which are gapped upon in- cluding SOC, each leaving behind a pair of Weyl nodes [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Angle-resolved photoemission spectroscopy (ARPES) stud- ies confirmed the existence of Fermi arcs in this system [35] and chiral edge modes were found in scanning tunneling mi- croscopy (STM) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The Berry curvature accumulated by the anticrossing line is claimed to be the source of large anoma- lous Hall and Nernst effects in this material [23, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The magnitude of the former reaches as high as 1200Ω−1cm−1 and is therefore comparable to the AHE in the related compound Fe3Sn2 [37, 38], in which, as demonstrated recently, only a fraction of the intrinsic AHE can be attributed to twisted nodal lines [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In addition, although the DFT band structure of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='13726v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='str-el] 31 Jan 2023 2 Co3Sn2S2 is relatively simple close to the Fermi level, it is to date unclear which band structure feature dominates the AHE: the gapped nodal loop or the Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Here, we address this fundamental question and determine the full optical conductivity tensor of Co3Sn2S2 by polarized infrared reflectivity and magneto-optical Kerr effect (MOKE) spectroscopy (For simplicity and readability, we omit the ex- plicit frequency dependence for the rest of the paper): ˆσ(ω) = � � σxx(ω) σxy(ω) 0 −σxy(ω) σxx(ω) 0 0 0 σzz(ω) � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' (1) Our results indicate that linear dichroism, namely the ratio of the conductivity in the kagome plane, σxx and out-of-plane, σzz, is a sensitive probe of topological features of layered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We find a resonant enhancement of the linear dichroism due to transitions along the gapped nodal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We reveal a giant magneto-optical optical activity in the same energy range by extending MOKE spectroscopy down to ¯hω = 25 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Specif- ically, we observe 1) a resonance peak at 40 meV in the Hall conductivity spectrum, σxy, which has not been detected be- fore, and 2) directly capture the fingerprints of the nodal loop in the optical conductivity without any extrapolation of σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Complemented by ab-initio calculations, we analyse the mo- mentum space distribution of the Hall spectral weight, which allows us to disentangle the contributions of the gapped nodal line and Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Polarized reflectivity spectra were measured on polished ab and ac surfaces of single crystals with a lateral size of ∼5 mm and ∼3 mm, respectively over the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='01 – 3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The di- agonal optical conductivity spectra were obtained by Kramers- Kronig-transformation of the reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The magneto-optical Kerr rotation, θ and ellipticity, η were measured at near nor- mal incidence on the same ab-cut crystal in ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 T and were antisymmetrized with respect to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The Hall conduc- tivity spectra were calculated using the complex Kerr rotation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='025 – 3 eV) according to θ +iη = − σxy σxx � 1+i 1 ε0ω σxx , (2) where ω is the angular frequency of the photon and ε0 is the vacuum permittivity [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The reflectivity and MOKE spectra measured between 10- 200 K are included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The out-of-plane reflectivity remarkably differs from the in-plane spectrum, the latter being in agreement with earlier reports [32, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In the overlapping energy range, the MOKE spectra agree with those published in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Importantly, our Kerr rotation and ellipticity obey the Kramers-Kronig relation and fulfil the magneto-optical sum rule, requiring that both parameters approach 0 for ω → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We show the low-energy spectrum for each independent component of the optical conductivity tensor in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1 (for a broad energy range see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The corresponding static conductivity values are shown for comparison and agree well with the respective spectra at the low-energy cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In panel Co Sn S 0 20 40 60 Re xx (102 1 cm 1) (a) 0 20 40 60 Re zz (102 1 cm 1) (b) 0 5 10 15 20 Im xy (102 1 cm 1) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 Energy (eV) 0 5 10 15 20 Re xy (102 1 cm 1) (d) Theory 10 K 20 K 40 K 60 K 80 K 100 K 120 K 140 K 160 K 170 K 172 K 174 K 176 K 178 K 180 K 200 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Comparison of the experimental conductivity spectra mea- sured between 10 and 200 K (colored lines) and the theoretical DFT spectra (black lines) calculated as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' a), b), c) and d) respectively show the real parts of the diagonal, Re σxx & Re σzz, as well as, the imaginary and real part of the off-diagonal conductivity spectra, Im σxy and Re σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' For comparison, the static conductivity values are plotted as colored squares at zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' (a), the real part of σxx exhibits a Drude-like increase towards zero energy responsible for the static conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' At 30 meV, a peak is forming below 100 K, separated well from the free carrier response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' For even higher energies, we find a small temperature dependent hump around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='25 eV and a step edge around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='6 eV before the conductivity becomes flat without significant temperature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' These features agree with earlier reports [32, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The out-of-plane conductivity spectrum, σzz in panel (b), strongly deviates from σxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Most strikingly, no sign of a Drude peak is observed down to our low-enegy cutoff, and the dc 3 conductivity is also much lower for this direction, indicated by the colored points at zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Therefore, we suspect that within the kagome planes, the strong orbital overlap between Co-sites can produce a coherent conduction, manifested in the Drude term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In contrast, the transport is likely due to incoherent hopping perpendicular to the planes [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' At 200 K, we find a peak at 40 meV, which shifts to smaller energies upon lowering the temperature until it eventually splits in two below 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' At higher energies, we find a minimum at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 eV, which becomes sharper at low temperatures, and a step edge at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' For even higher energies, σzz is featureless though slightly increasing without distinct temperature dependence, similar to σxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1(c) & (d), the Hall conductivity spectrum σxy shows a strong resonance at 40 meV, in coincidence with the in- and out-of-plane diagonal components and their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We empha- size that this far-infrared range has not been covered so far, while the higher energy part of the spectra agree very well with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Both its imaginary and real parts exhibit a large enhancement towards low temperatures, where the peak in the real part overshoots the dc-AHE below 60 K with a magni- tude as high as 2000 Ω−1cm−1 at 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The good agreement between the low-energy tail of the real part of σxy and the dc- AHE together with the formerly published featureless THz data [42], suggests that there are no further excitations in the narrow uncovered energy interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Furthermore, as the scattering rate obtained from the Drude peak for σxx is below the cutoff for σxy, we conclude that the giant anomalous Hall conductivity of Co3Sn2S2 has dominantly intrinsic origin and it is generated by the interband resonance observed here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In order to reveal the microscopic origin of the observed spectral features, we performed ab–initio calculations provid- ing all symmetry allowed elements of the conductivity tensor [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The theoretical spectra are coplotted with the experiment in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Disregarding the intraband contribution which is not included in the theory, the spectra qualitatively reproduce all conductivity components on a large energy scale, although the spectral features appear slightly shifted to higher energies compared to experiment, which may indicate correla- tion effects [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We quantified the linear dichroism for in- and out-of-plane polarizations by calculating Re σzz/Re σxx from the spectra shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' As demonstrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 2, the theory agrees with the experiment remarkably well, especially at higher tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We find that around the resonance at 40 meV, σzz becomes three times larger than σxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' As the temperature is decreased in the experiment, some of the spectral weight splits and moves to lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Although this is not properly captured by the DFT calculations the overall tendency, that σzz is stronger at the resonances, remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The Hall conductivity is also well reproduced by the the- ory as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1 and S2(c) & (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The imaginary part shows a sudden increase at 40 meV, where the resonance is observed in the experiment, though it is not that pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Importantly, the theory properly captures the 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='9 meV peak in Reσxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The experimental feature is somewhat sharper like in the case of the imaginary part, which may again be related to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 Energy (eV) 0 1 2 3 4 Re zz/Re xx 0 1 2 3 0 2 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Optical anisotropy spectra Re σzz/Re σxx with the same colorcode as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The inset shows the spectra over a broader energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' electronic correlations [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The dc–extrapolation of the theory yields the same AHE as the magnetotransport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Backed with this remarkable agreement, we now analyse the band structure origin of this giant optical Hall effect signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We deduced the momentum decomposition of the off- diagonal conductivity by introducing the Hall spectral weight Hxy(ω,k) as σxy(ω) = ie2/¯hV � BZ Hxy(ω,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' (For details see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S1 and text of SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 3(b) shows the Hall spectral weight of the 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='9 meV low-energy peak in Re σxy on one of the mir- ror planes containing the nodal line and the Weyl points as indicated by the rainbow line and black labels, respectively [34, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The black line is the border of the BZ, the green lines plot the Fermi surface, and the colormap presents the spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' For better orientation, panel (a) plots the nodal lines where the coloring encodes the position of the band crossing relative to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We find several hot spots of Hall conductivity on the slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Interestingly, there is no significant contribution from the Weyl points as they are located 60 meV above the Fermi energy and cannot contribute to the response in the range below [34, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Instead, we find regions close to Γ and in the vicinity of A with positive and negative weights right next to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Since the conductivity is determined by integration over the BZ, the opposite weight from these points cancel to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' This leaves the large positive patch on the Γ − A line giving the dominant contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Aside from the regions around Γ, we always observe a hot spot when the nodal line comes close to or crosses the Fermi energy (light green segments of the loop), hence the hotspots always connect two close regions of the Fermi surface (dark green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Although the hot spots seem to have a similar origin, they behave differently in producing optical weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In order to see the underlying shape of the bands, we introduce two non-high- symmetry points A and B, to plot the band structure along cuts through the hot spots, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 3(c) with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The grey shading indicates the relevant low-energy range and is shaded in the same fashion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Along the B−A line, we cut through a hot spot with subsequent positive and negative 4 L A B Γ L A (c) Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 SOC Spin Up Spin Dn 140 70 0 70 140 Energy (meV) (a) kx kz ky T U L Γ kz Γ T U L A B Weyl Weyl Hxy(ω,k) (102 Å2) 0 5 10 10 5 ky (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' (a) Location of the nodal lines in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The colorscale encodes the position relative to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' (b) The Hall spectral weight of the calculated 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='9 meV peak on the mirror plane containing the nodal line and Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' (c) Bandstructure along the main contributing areas of the real part of the off-diagonal optical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The grey shading highlights the energy range below 50 meV where we expect contributions to the peak in Re σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The non-high-symmetry points A and B are (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='301) and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='7373, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0) in reciprocal lattice units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Here, the two red spin up bands forming the nodal line are close to the Fermi energy, so with SOC, one of the bands is filled and the other is empty in between the crossings, allowing the optical transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Importantly, the tilt of the crossing points is opposite, which is the reason for the different signs of their contribution to σxy [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Around Γ, we have a similar situation where two red spin up band inversions appear with opposite tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The SOC opens a gap, which allows transitions once the upper band is above the Fermi energy, also producing an optical Hall response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The situation must be different along the A−Γ line, where the large positive hot spot does not have a negative partner nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Here, we observe only one strongly tilted crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Again, with SOC the upper band is pushed above the Fermi level enabling the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Due to the strong tilt, the two bands stay nearly parallel over a relatively broad k-interval, so the Berry curvature of the gapped nodal line is summed up over a narrow energy range which produces the large patch of optical weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' By inversion and 3–fold rotation symmetry, we expect a total of 6 such spots in the BZ dominating the anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Since the interband transition from this feature also gives rise to diagonal conductivity, we directly compare σxx and σxy by calculating the Hall angle, ΘH = arctanRe(σxy/σxx), which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' At 40 meV, the giant off–diagonal conductivity has almost the same magnitude as the diagonal conductivity producing a very large Hall angle of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' When the Kubo formula is written using the circular momentum operators Im σxy = e2π 4m2V ¯h ∑ k,n,n′ |f(εn(k))− f(ε′ n(k))| ωnn′ δ(ω −ωnn′) × ���⟨n,k|p+|n′,k⟩ ��2 − ��⟨n,k|p−|n′,k⟩ ��2� , (3) σxy depends on the difference of the two circular components, whereas σxx is given by their sum [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' As a consequence, the largest possible Hall angle for an interband transition is 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' This is realized when one matrix element is zero, which indicates an almost fully polarised transition in the present case, yielding a nodal line resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In summary, we provide a showcase for disentangling the contributions of various topological features to the AHE, and identify the gapped nodal line as the main source of the giant AHE in Co3Sn2S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Facilitated by far infrared MOKE spec- troscopy, we observe a low-energy magneto-optical resonance for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' As the zero energy extrapolation of this interband transition explains the static Hall conductivity, we confirmed that the AHE is dominantly intrinsic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Our ab-initio calculations are able to reproduce the experimental spectra with remarkable accuracy, allowing a momentum and band de- composition of the optical Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' We find that the Weyl points located 60 meV above the Fermi energy only yield singular contributions in a small k-volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' By contrast, the nodal line segments approaching or crossing the Fermi energy produce large AHE hotspots after being gapped by SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In ad- dition, we verify that the tilt of the nodal line is a crucial factor, which can lead to pairwise cancellation or in contrast, produce a nodal line resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Remarkably, the linear dichroism is significantly enhanced by the nodal line resonance, leading to a potentially new signature of topological states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Finally, we note that this magneto-optical analysis is applicable for any material where large AHE is suspected from topological bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Since in magnetic materials, the electronic topology may cou- ple to the magnetic order, it is also a suitable tool to monitor the effects of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' external fields for manipulating topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The authors are grateful to Christine Kuntscher, Liviu Chion- cel, and Artem Pronin for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' This work was supported by the Hungarian National Research, Development 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='14 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='07 500 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='07 ←-500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='14 1000 kx10 5 0 5 10 kx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='14 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='07 500 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='07 ←-500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='14 1000 kx5 and Innovation Office – NKFIH grants FK 135003 and Bolyai 00318/20/11 and by the Ministry of Innovation and Technol- ogy and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Sándor Bordács is supported by the ÚNKP-22- 5-BME-280 new national excellence program of the ministry for innovation and technology from the source of the national research, development and innovation 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+page_content=' Marchand, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Marrazzo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Mokrousov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Mustafa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Nohara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Nomura, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Paulatto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Poncé, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Ponweiser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Qiao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Thöle, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Tsirkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Wierzbowska, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Marzari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Vanderbilt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Souza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Mostofi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Yates, Wannier90 as a community code: new features and applications, Journal of Physics: Condensed Matter 32, 165902 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' [61] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Jiang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Guo, Large magneto-optical effect and magnetic anisotropy energy in two-dimensional metallic ferro- magnet Fe3GeTe2, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' B 105, 014437 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 8 SUPPLEMENTAL MATERIAL Crystal growth for the magneto-optical measurements A large single crystal of Co3Sn2S2 was grown in a cylindro- conical shape (1 cm in diameter and 5 cm in length) from its melt by a modified Bridgman method [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' About 10 g of polycrystalline Co3Sn2S2 synthesized by a solid state reaction was charged in a tipped glassy carbon crucible which was sealed under vacuum in a quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The sealed ampule was suspended by a Kanthal thread from the top to the hot zone of a vertical tube furnace and heated over 30 hours up to 1000◦C, kept for 6 h, and then slowly cooled over 72 h to 800◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' After furnace cooling, a crystal has been removed from the crucible and could be easily cleaved in the (001) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Crushed parts were investigated by powder x-ray diffraction (XRD), Laue x-ray spectroscopy and wave-length dispersive x-ray spectroscopy (WDX), those indicated a single-phase and high-quality grown crystal with stoichiometric chemical composition of Co3Sn2S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Crystal growth for magnetotransport Single crystals of Co3Sn2S2 were grown, in a flat hexagonal shape, out of Sn flux as published elsewhere [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Lumps of Co (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='9 % high purity chemicals) and grains of Sn (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='999 % high purity chemicals) were mixed with S powder (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='999 % nacalai tesque) in a molar ratio of Co : S: Sn = 8: 6: 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' A mixture with a total mass of ≈ 15g was charged in Al2O3 growth crucible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The growth crucible was covered by another inverted Al2O3 crucible filled with quartz wool and both were sealed under vacuum in a quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The ampule was fired in a muffle furnace at 1050◦C for 6 hours and then slowly cooled to 700◦C over 72 hours at which the ampule was removed from the furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The flux was removed from the crystals via a rapid decanting of the ampule followed by a subsequent spinning in a centrifuge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The grown crystals were characterized by powder XRD and WDX and the crystal orientations were determined by Laue x-ray spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Reflectivity measurements The reflectivity spectra of Co3Sn2S2 were obtained in a Bruker IFS/66 FTIR-spectrometer for the MIR-VIS range and a Bruker Vertex 80v for the FIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The spectra were measured in the frequency range 80-32000 cm−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='01-4 eV) from room temperature down to 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' As references, a silver and gold mirror where used in the MIR-VIS and FIR experiments, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The optical conductivity σ1 was calculated by using Kramers-Kronig analysis on the merged spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' At this point, the low-energy side of the reflectivity spectrum was extrapo- lated by using a Hagen-Rubens law and the dc-conductivity, while for the UV the reflectivity spectrum was extrapolated with free electron behaviour setting in at 106 cm−1 and an exponent for the interband regime of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' MOKE-spectroscopy The broadband MOKE spectra were recorded in near-normal incidence and were combined from several measurements in different frequency ranges, employing grating and interfer- ometer based spectrometers as described elsewhere [39, 52– 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Small permanent magnets provided a field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 T at the sample position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Because of the large uniaxial anisotropy in this material, the sample was warmed up over Tc between measurements with reversed field direction to ensure proper antisymmetrisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In the overlapping energy range, the MOKE spectra agree with those published in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Beside the giant Kerr rotation with a peak magnitude of -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 deg at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='09 eV, we resolve a peak around 50 meV in the ellipticity with a magnitude of 2 deg, which was not detected before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' DFT calculations The electronic structure of Co3Sn2S2 is calculated using VASP code [55–57] based on the density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The generalized gradient approximation of Perdew-Burke- Ernzerhof was adopted for the exchange-correlation functional [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The crystal structure for Co3Sn2S2 is of trigonal form with a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='379 Å and α = 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='8658◦, which is the experimentally determined lattice constant [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' In the self-consistent band structure calculations, Γ-centered k meshes of 24 × 24 × 24 were used in the Brillouin zone integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The optical prop- erties are further evaluated using the Wannier functions, and the Kubo-Greenwood formula [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Below, we present the formula used for calculating Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1(c) and deducing the Hall spectral weight of a certain transition energy plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' σk,αβ(¯hω) = ie2 ¯hV ∑ k,n,n′ ( fn′,k − fn,k)Re � εn′,k −εn,k εn′,k −εn,k −(¯hω +iη) � Ann′,α(k)Ann′,β(k) = ie2 ¯hV ∑ k Hαβ(ω,k) (S1) The α and β are the indices in Cartesian coordinates, V is the cell volume, and fn,k = f(εn,k) is the Fermi-Dirac distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' ω is the optical frequency and η > 0 is the smearing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Finally, Ann′,α = � un,k ��i∇kα ��un′,k � is the Berry connection and Hαβ is the Hall spectral weight at certain transition en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Importantly, if we apply ⟨v⟩ = ⟨p⟩/m and the relation � ψn,k ��v ��ψn′,k � = −i/¯h · (εn′,k − εn,k)Ann′(k), we would arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 3 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' If we take ω and η as zero, we obtain the ordinary Berry curvature for the anomalous Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 9 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 Reflectivityxx (a) Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 Reflectivityzz (b) Energy (eV) 3 2 1 0 1 Rotation (deg) (c) 10 K 20 K 40 K 60 K 80 K 100 K 120 K 140 K 160 K 170 K 172 K 174 K 176 K 178 K 180 K 200 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 Energy (eV) 2 1 0 1 2 Ellipticity (deg) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Spectra measured for several temperatures between 10 and 200 K of a) the reflectivity on the kagome plane and b) along the stacking direction c) Kerr-rotation and d) ellipticity on the ab-plane in the energy range up to 1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' A fine mesh of 100 × 100 × 100 k points are applied during the integration with good convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The Wannier functions were constructed using Co d, Sn s, p, and S s, p orbitals with a resulting tight-binding model well describing the DFT band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Importantly, from the main text we notice that the experimental low-frequency peak of the real part off-diagonal optical conductivity Reσxy is nicely captured by the theoretical calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 10 0 20 40 60 Re xx (102 1 cm 1) (a) 0 20 40 60 Re zz (102 1 cm 1) (b) 0 5 10 15 20 Im xy (102 1 cm 1) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 Energy (eV) 5 0 5 10 15 Re xy (102 1 cm 1) (d) 10 K 200 K Theory FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Conductivity spectra over a broad energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 11 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' a) Calculated band structures along the high symmetry line with and without the spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' The eigenvalues of the Wannier tight-binding model are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' b) Comparison of band structures over a large energy range calculated by density functional theory and Wannier tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 Energy (eV) 0 10 20 30 40 H (deg) Theory 10 K 20 K 40 K 60 K 80 K 100 K 120 K 140 K 160 K 170 K 172 K 174 K 176 K 178 K 180 K 200 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' Hall angle spectra arctanRe(σxy/σxx) with a maximum of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content=' 6 5 DFT WANNIER 3 Energy (eV 2 A B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 Spin up 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='1 Spin Dn SOC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='2 annier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} +page_content='5 W U' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FST4oBgHgl3EQfHjj7/content/2301.13726v1.pdf'} diff --git a/ktE9T4oBgHgl3EQfVheR/content/tmp_files/2301.07334v1.pdf.txt b/ktE9T4oBgHgl3EQfVheR/content/tmp_files/2301.07334v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ffabd1dc070cc65b22d6c27c619efde5f20dcde7 --- /dev/null +++ b/ktE9T4oBgHgl3EQfVheR/content/tmp_files/2301.07334v1.pdf.txt @@ -0,0 +1,872 @@ +arXiv:2301.07334v1 [math.NT] 18 Jan 2023 +Almost Repdigits in k−generalized Lucas +Sequences +Alaa ALTASSAN and Murat ALAN +King Abdulaziz University +Department of Mathematics, Jeddah, 21589, Saudi Arabia +Yildiz Technical University +Department of Mathematics, Istanbul, 34210, Turkey +January 19, 2023 +Abstract +Let k ≥ 2 and (L(k) +n )n≥2−k be the k−generalized Lucas sequence with +initial condition L(k) +2−k = · · · = L(k) +−1 = 0, L(k, +0 += 2, L(k) +1 += 1 and each +term afterwards is the sum of the k preceding terms. A positive integer +is an almost repdigit if its digits are all equal except for at most one +digit. In this paper, we work on the problem of determining all terms of +k−generalized Lucas sequences which are almost repdigits. In particular, +we find all k−generalized Lucas numbers which are powers of 10 as a +special case of almost repdigits. +Key Words: k−Lucas numbers, repdigits, almost repdigits, linear forms +in logarithms +2010 Mathematics Subject Classification: 11B39, 11J86, 11D61. +1 +Introduction +Let k ≥ 2 be an integer. The k−generalized Lucas sequence or, for simplic- +ity, the k−Lucas sequence is a sequence given by the recurrence relation +L(k) +n += L(k) +n−1 + · · · + L(k) +n−k +for all +n ≥ 2, +with the initial values L(k) +i += 0 for i = 2 − k, . . . , −1, L(k) +0 += 2 and L(k) +1 += +1. We call the terms of this sequence k−Lucas numbers for simplicity. +For k = 2, this sequence is the classical Lucas sequence, and hence the +k−Lucas sequence is a generalization of the Lucas sequence from binary +recurrence sequence to the order k recurrence sequence. +Recall that, a positive integer whose all digits are equal is called a +repdigit. In recent years, many researches have been performed to find all +terms of some sequences related to repdigits, see for example [1, 4, 5, 9, +10, 11, 12, 19, 20, 24, 25, 26]. In this study, we search the numbers similar +to the repdigits in k−Lucas numbers. +1 + +In [14] and [18], all square and perfect power positive integers whose +digits are all equal except for one digit have been examined without giving +a specific name. We call a positive integer an almost repdigit if its all digits +are equal except for at most one digit. This numbers can be written of +the form +a +�10d1 − 1 +9 +� ++ (b − a)10d2, +0 ≤ d2 < d1 +and +0 ≤ a, b ≤ 9. +Usual repdigits and the numbers of the form b10d2 are two particular +cases of almost repdigits which corresponds to the cases a = 0 and b = a, +respectively. Thus, almost repdigits are a generalization of repdigits. +Recently, in [2], the authors found all k−generalized Fibonacci num- +bers that are almost repdigits. In this paper, we continue to search almost +repdigits by taking into account k−generalized Lucas numbers as an ana- +logue of the study in [2]. In other words, we consider the Diophantine +equation +L(k) +n += a +�10d1 − 1 +9 +� ++(b−a)10d2, +0 ≤ d2 < d1 +and +0 ≤ a, b ≤ 9 (1) +in non negative integers d1, d2, a and b and we prove the following theorem. +Theorem 1 The Diophantine equation (1) has solutions only in the cases +L(2) +11 = 199, L(2) +12 = 322, L(3) +8 += 118, L(3) +10 = 399, L(7) +10 = 755 and L(9) +10 = +766, when L(k) +n +has at least three digits. +Since, the numbers having at most two digits are trivially almost +repdigits, we state the above theorem only for L(k) +n +which are consist +of at least three digits. Thus, from now on we take d1 ≥ 3 and hence +n > 5. +The proof of the above theorem, mainly depends on two effective meth- +ods, that is, the linear forms in logarithms of algebraic numbers due to +Matveev [21] as well as reduction algorithm due to Dujella and Peth˝o +[13], which is in fact originally introduced by Baker and Davenport in [3]. +We give some details of these methods in the next section whereas in the +third section, we give the main properties of k−Lucas sequences that we +will need later. We devoted the forth section to the proof of Theorem 1. +It is also worth to note that, we implemented the software Maple for all +calculations and computations in the proof the Theorem. +2 +The Tools +Let θ be an algebraic number, and let +c0xd + c1xd−1 + · · · + cd = c0 +d +� +i=1 +(x − θ(i)) +be its minimal polynomial over Z, with degree d, where the ci’s are rela- +tively prime integers with c0 > 0, and the θ(i)’s are conjugates of θ. +2 + +The logarithmic height of θ is defined by +h(θ) = 1 +d +� +log c0 + +d +� +i=1 +log +� +max{|θ(i)|, 1} +�� +. +If θ = r/s is a rational number with relatively prime integers r and s and +s > 0, then h(r/s) = log max{|r|, s}. The following properties are very +useful in calculation of a logarithmic height : +• h(θ1 ± θ2) ≤ h(θ1) + h(θ2) + log 2. +• h(θ1θ±1 +2 ) ≤ h(θ1) + h(θ2). +• h(θs) = |s|h(θ), s ∈ Z. +Theorem 2 (Matveev’s Theorem) Assume that α1, . . . , αt are posi- +tive real algebraic numbers in a real algebraic number field K of degree dK +and let b1, . . . , bt be rational integers, such that +Λ := αb1 +1 · · · αbt +t − 1, +is not zero. Then +|Λ| > exp +� +K(t)d2 +K(1 + log dK)(1 + log B)A1 · · · At +� +, +where +K(t) := −1.4 · 30t+3 · t4.5 +and +B ≥ max{|b1|, . . . , |bt|}, +and +Ai ≥ max{dKh(αi), |log αi|, 0.16}, +for all +i = 1, . . . , t. +For a real number θ, we put ||θ|| = min{|θ − n| : n ∈ Z}, which represents +the distance from θ to the nearest integer. Now, we cite the following +lemma which we will use it to reduce some upper bounds on the variables. +Lemma 3 [6, Lemma 1] Let M be a positive integer, and let p/q be a +convergent of the continued fraction of the irrational γ such that q > 6M. +Let A, B, µ be some real numbers with A > 0 and B > 1. If ǫ := ||µq|| − +M||γq|| > 0, then there is no solution to the inequality +0 < |uγ − v + µ| < AB−w, +in positive integers u, v and w with +u ≤ M +and +w ≥ log(Aq/ǫ) +log B +. +We cite the following lemma from [15, Lemma 7]. +Lemma 4 Let m ≥ 1 and T > (4m2)m. Then we have +x +(log x)m < T ⇒ x < 2mT (log(T ))m. +3 + +3 +Properties of k−Fibonacci and k−Lucas +Numbers +The characteristic polynomial of k−Lucas numbers is +Ψk(x) = xk − xk−1 − · · · − x − 1 +which is an irreducible polynomial over Q[x]. The polynomial Ψk(x) has +exactly one real distinguished root α(k) outside the unit circle [22, 23, 27]. +The other roots of Ψk(x) are strictly inside the unit circle [23]. This root +α(k), say α for simplicity, placed in the interval +2(1 − 2−k) < α < 2 +for all +k ≥ 2. +Let +fk(x) = +x − 1 +2 + (k + 1)(x − 2). +(2) +It is known that the inequalities +1/2 < fk(α) < 3/4 +and +|fk(αi)| < 1, +2 ≤ i ≤ k +(3) +are hold, where α := α1, · · · , αk are all the roots of Ψk(x) [6, Lemma 2]. +In particular, we deduce that fk(α) is not an algebraic integer. In the +same Lemma, it is also proved that +h(fk(α)) < 3 log(k) +holds +∀k ≥ 2. +(4) +In [10], Bravo and Luca showed that +L(k) +n += +k +� +i=1 +(2αi − 1)fk(α)(αi)n−1 +and +���L(k) +n +− (2α − 1)fk(α) +��� < 3/2 +(5) +for all k ≥ 2. As in the classical k = 2 case, we have the similar bounds as +αn−1 ≤ L(k) +n +≤ 2αn +(6) +for all n ≥ 1 and k ≥ 2 [10]. +4 +Proof of Theorem 1 +Assume that Equation (1) holds. By combining the inequality +10d1−2 < a +�10d1 − 1 +9 +� ++ (b − a)10d2 ≤ 2 · 10d1, +and (6), we get that +d1 < log 2 +log 10(n + 1) + 2 < 0.31n + 2.31 < n − 1 +(7) +and +0.2n − 0.6 < log((1 + +� +(5))/2) +log 10 +(n − 1) − log 2 +log 10 < d1 +(8) +for all n > 5. +First, we assume that a ̸= 0. We examine the case a = 0 in the end of +this section. +4 + +4.1 +The Case n < k + 1 and Almost Repdigits of +the Form 3 · 2n +Assume that n ≤ k. In this case, L(k) +n += 3 · 2n−2, and hence Equation (1) +can be written as +27 · 2n−2 = a +� +10d1 − 1 +� ++ 9(b − a)10d2. +Taking modulo 2d2 and modulo 2d1, we find that d2 ≤ 3 and d1 ≤ 13. +Hence, from (8), we see that n < 70. A quick calculation shows that +when n < 70, there is no almost repdigits of the form 3 · 2n−2 with at +least three digits. +So, from now on, we take n ≥ k + 1. +4.2 +A Bound for n Depending on k +By rewriting (1) as +L(k) +n ++ a/9 − (b − a)10d2 = a10d1/9, +and by using (5), we get +���fk(α)(2α − 1)αn−1 − a10d1/9 +��� ≤ (3/2) + +���a/9 − (b − a)10d2 +��� . +Therefore, we obtain +|Λ1| ≤ 27/2 +10d1 + +1 +10d1 + |b − a|(9/a) +10d1−d2 +≤ +87 +10d1−d2 , +(9) +where +Λ1 := αn−110−d1fk(α)(2α − 1)9/a − 1. +We take +η1 := α, η2 := 10, η3 := fk(α)(2α − 1)9/a, +b1 := n − 1, b2 := −d1, b3 := 1. +Note that from (4), we find h(η3) ≤ h(9/a) + h(fk(α)) + h(2α − 1) < +8 log(k), since h(2α − 1) < log 3 [10, page 147]. +We have also Λ1 ̸= 0. Indeed, if Λ1 = 0, then we would get +a10d1/9 = fk(α)(2α − 1)αn−1. +Conjugating both sides of this relation by any one of the automorphisms +σi : α → αi for any i ≥ 2 and by taking the absolute values, we find that +100 < |fk(αi)||2α − 1||αi|n−1 < 3, +a contradiction. Thus,Λ1 ̸= 0. Other calculations are doneby using similar +techniques as in the k−Fibonacci case. So, by combining the result of +Theorem 2 and the fact that log (Λ1) < log 87−(d1 − d2) log 10, we obtain +d1 − d2 < 4.8 · 1012 · k4(log2 k) log (n − 1). +(10) +5 + +By rearranging Equation (1) as follows +L(k) +n ++ a/9 = a10d1/9 + (b − a)10d2, +and using (5), we get +|Λ2| ≤ 5 +2 +1 +fk(α)αn−1 ≤ +5 +2αn−1 . +(11) +where Λ2 := α−(n−1)10d1fk(α)−1(2α − 1)−1((a/9) + (b − a)10d2−d1) − 1. +By the similar argument as above we see that Λ2 ̸= 0. Let +η1 := α, η2 := 10, η3 := fk(α)−1(2α − 1)−1((a/9) + (b − a)10d2−d1) +with b1 := −(n − 1), b2 := d1, b3 := 1. All η1, η2 and η3 belong to the real +number field K = Q(α) and therefore we take dK = 2, to be the degree of +the number field K. Using the properties of logarithmic height, and the +fact that h(2α − 1) < log 3, we find +h(η3) ≤ h(fk(α)−1) + h((2α − 1)−1) + h((a/9) + (b − a)10d2−d1) +≤ 3 log(k) + h((2α − 1)) + h(a/9) + h(b − a) + h(10d2−d1) + log(2) +≤ 3 log(k) + log 3 + log(144) + |d2 − d1| log(10) +< 12 log(k) + |d2 − d1| log(10). +By Theorem 2, we get a bound for log(Λ2). Then by combining this +bound with the one comes from (11), we get +n − 1 < 2.4 · 1025k8(log(k))3(log(n − 1))2. +Therefore, from Lemma 4, we have +n < 1.3 · 1030k8(log(k))5. +(12) +4.3 +The Case k ≤ 470 +Let +Γ1 := (n − 1) log α − d1 log 10 + log(fk(α)(2α − 1) · 9/a). +Then +Λ1 := |exp(Γ1) − 1| < 87/10d1−d2. +We claim that d1 − d2 < 65. Suppose that d1 − d2 > 3. Then 87/10d1−d2 < +1/2 and therefore |Γ1| < 174/10d1−d2. So we have +0 < +����(n − 1) log α +log 10 − d1 + log(fk(α) · (2α − 1) · 9/a) +log 10 +���� < 174/10d1−d2 log 10. +(13) +For each 2 ≤ k ≤ 470, we take Mk := 1.3 · 1030k8(log(k))5 > n and +τk = log α +log 10. Then, for each k, we find a convergent pi/qi of the continued +fraction of irrational τk such that qi > 6Mk +After that, we calculate ǫ(k,a) := ||µ(k,a)qi|| − Mk||τkqi|| for each a ∈ +{1, 2, . . . , 9}, where +µ(k,a) := log(fk(α) · (2α − 1) · 9/a) +log 10 +. +6 + +If ǫ(k,a) < 0, then we repeat the same calculation for qi+1. Except for +(k, a) = (2, 9). In fact 0.00008 < ǫ(k,a). Thus, from Lemma 3, we find an +upper bound on d1 − d2 for each 2 ≤ k ≤ 470 such that none of them are +greater than 61. So, we conclude that d1 − d2 < 65 as we claimed. +If (k, a) = (2, 9), then τ = µ(k,a) and hence ǫ(k,a) = 0. So, in this case, +(13) is +0 < +����n log α +log 10 − d1 +���� < 174/10d1−d2 log 10. +In other words, +0 < +���� +log α +log 10 − d1 +n +���� < 174/10d1−d2 log 10. +(14) +From, the theory of continued fractions, we see that this implies that the +bound d1 − d2 < 65 is valid. +Next, let +Γ2 := −(n−1) log α+d1 log 10+log(fk(α)−1·(2α−1)−1·((a/9)+(b−a)10d2−d1)). +Thus, +Λ2 := |exp(Γ2) − 1| < 5/2αn−1 < 1/2. +Hence, we get that +0 < +����(n − 1) log α +log 10 − d1 + µ(k,d1−d2,a,b) +���� < +5 +αn−1 log 10. +(15) +where +µ(k,d1−d2,a,b) := −log(fk(α)−1 · (2α − 1)−1 · ((a/9) + (b − a)10d2−d1)) +log 10 +. +This time we calculate ǫ(k,d1−d2,a,b) := ||µ(k,d1−d2,a,b)qi|| − Mk||τkqi|| +for each d1 − d2 ∈ {1, 2, . . . , 65}, a ∈ {1, 2, . . . , 9} and b ∈ {0, 1, . . . , 9}. +We apply Lemma 3 to Equation (15), and therefore we find an upper +bound on n−1 for each 2 ≤ k ≤ 470, say nL(k). For example nL(3) < 150, +nL(10) < 147, nL(100) < 178, nL(200) < 197, nL(300) < 296, nL(400) < +396 and n(470) < 465 are some of these bounds. +By writing a short computer programme in Maple, and using the +obtained bounds, we find that L(2) +11 += 199, L(2) +12 += 322, L(3) +8 += 118, +L(3) +10 += 399, L(7) +10 += 755 and L(9) +10 += 766, are the only k−Lucas num- +bers which are almost repdigits with at least three digits, as we claimed +in Theorem 1. Now, we turn our focus to the case k > 470. +4.4 +The Case k > 470 +We use the following lemma [24, Lemma 2.6]. +Lemma 5 If n < 2k/2, then the following estimates hold: +L(k) +n += 3 · 2n−2(1 + ζ(n, k)), +where +|ζ(n, k)| < +1 +2k/2 . +7 + +For k > 470, the inequality +n < 1.3 · 1030k8(log(k))5 < 2k/2, +holds and hence from the above Lemma, we have that +���3 · 2n−2 − L(k) +n +��� < 3 · 2n−2 +2k/2 . +(16) +Now, we turn back to (1) one more time to rewrite it as +���L(k) +n +− (a/9)10d1 +��� < (a/9) + |b − a|10d2. +(17) +Thus, by combining (16) and (17), we get +���3 · 2n−2 − (a/9)10d1 +��� < 3 · 2n−2 +2k/2 + (a/9) + |b − a|10d2. +Therefore, we have +Λ3 := +���2n−210−d127/a − 1 +��� < 1 +2λ , +(18) +where λ := min{(k/2) − 5, (d1 − d2)log(10) +log(2) − 8}. +Let η1 := 2, η2 := 10, η3 := 27/a and b1 := n − 2, b2 := −d1, b3 := 1. +Applying Theorem 2 to Λ3, we get λ < 2.2 · 1012 log n +where we used +log n < log(1.3 · 1030k8(log(k))5) +< log(4.3) + 3 log(10) + 8 log k + 5 log log(k) +< 50 log k. +Thus, if λ := (k/2) − 5, then we get a bound for k as +k < 1016. +If λ := (d1 − d2)log(10) +log(2) − 8, then we get +d1 − d2 < 3.4 · 1013 log k. +(19) +This bound of d1 − d2 also leads to an upper bound of k. To do this, +we rewrite (1) as +���L(k) +n +− (a/9)10d1 − (b − a)10d2 +��� ≤ (a/9) ≤ 1. +(20) +By combining (20) with (16), we find that +Λ4 := +���2−(n−2)10d1((a/9) + (b − a)10d2−d1)(1/3) − 1 +��� < +1 +3 · 2n−2 +2 +2k/2 ≤ +2 +2k/2 . +We take +η1 := 2, η2 := 10, η3 := (1/3)((a/9) + (b − a)10d2−d1) +8 + +and b1 := −(n − 2), b2 := −d1, b3 := 1. Then +h(η3) = h(a/9) + h(b − a) + |d2 − d1|h(10) + h(3) + log 2 +< log 432 + (d1 − d2) log 10. +Other calculations are similar to those for Λ4 as K = Q, dK = 1, +B := n > n − 2. h(η1) = log 2, h(η2) = log 10. +Moreover Λ4 ̸= 0. Indeed, 2n−2 = 10d1(a/9) + (b − a)10d2 implies that +a = 9 and d2 = 0. For d1 = 3, clearly the equation 2n−2 = 10d1 + b − 9 +has no solution in integers. So d1 > 3. Thus, congruence consideration +modulo 24 shows that this equation has no integer solutions for 0 ≤ b ≤ 9. +Hence, Λ4 ̸= 0. +Therefore, Theorem 2 together with (4.4) give that +log 2−(k/2) log 2 > −1.4·306·34.5(1+log n) log 2·log 10·(log 432+(d1 − d2) log 10). +At this point, we use the upper bound of d1 − d2 which was given in (19), +and by using the estimates log 432 < log k and log n < 50 log k, we obtain +the desired upper bound for k as +k < 3 · 1031. +(21) +Thus, by (12), we have also a bound for n as +n < 1.8 · 10291. +(22) +4.5 +Reducing the Bound on k +We will reduce these highly large upper bounds. Let +Γ3 := (n − 2) log 2 − d1 log 10 + log(27/a). +(23) +Then Λ3 := |exp(Γ3) − 1| < 1 +2λ . We will find a feasible bound for λ. +Suppose that λ > 1. Then, 1 +2λ < 1 +2 and hence we get that |Γ3| < 2 +2λ . +In this case, we don’t need to consider the case a = 9 separately. +From (23), we write +0 < +����(n − 2) log 2 +log 10 − d1 + log(27/a) +log 10 +���� < +2 +2λ log 10. +(24) +Let M := 1.8 · 10291 > n and τ = log 2 +log 10. Then, the denominator of +the 588th convergent of τ, say q588, exceeds 6M. +Then +ǫa := ||µaq588|| − M||τq588|| > 0.029559, +for each a ∈ {1, 2, . . . , 9}, where +µa := log(27/a)/log 10. +Thus, by applying Lemma 3, we get λ < 975. +9 + +Hence, if λ = k/2−5, then k < 1960. Assume that λ = (d1 − d2)log(10) +log(2) − +8. Then +d1 − d2 < 296 < 300. +Let +Γ4 = +���(n − 2) log 2 − d1 log 10 − log((a/9) + (b − a)10d2−d1)(1/3) +��� +(25) +Then +Λ4 := |exp(Γ4) − 1| < +2 +2k/2 < 1 +2. +(26) +So +0 < | +Γ4 +log 10| < +����(n − 2) log 2 +log 10 − d1 + µ(a,b,d1−d2) +���� < +4 +2k/2log 10 +(27) +where +µ(a,b,d1−d2) := −log ((a/9) + (b − a)10d2−d1)(1/3) +log 10 +. +We take M := 1.8 · 10291 > n and τ = log 2 +log 10. +This time, we take q595, which is the denominator of the 595th con- +vergent of τ, as qi > 6M. +Let ǫ(a,b,d1−d2) := ||µ(a,b,d1−d2)q595||−M||τq595|| for each a ∈ {1, 2, . . . , 9}, +b ∈ {1, 2, . . . , 9} and d1 − d2 ∈ {1, 2, . . . , 300}. We find that 0.000036 < +ǫ(6,0,272) ≤ ǫ(a,b,d1−d2) for all a, b, d1 − d2 except for (a, b, d1 − d2) = +(9, 2, 1), (9, 5, 1), (9, 5, 2) since these three triples of (a, b, d1 − d2), we have +that µ(a,b,d1−d2) < 0. +Then, except for the above three triples, from Lemma 3, we conclude +that k < 2000. Hence, from (12), n < 8.5 · 1060. +If (a, b, d1 − d2) = (9, 2, 1), then +µ(9,2,1) := − +log (1 − 7 +10)(1/3) +log 10 += 1 ∈ Z. +So, in this case we may write +Γ4 = +��(n − 2) log 2 − d1 log 10 − log 10−1�� += |(n − 2) log 2 − (d1 − 1) log 10| , +and hence +0 < +���� +log 2 +log 10 − d1 − 1 +n − 2 +���� < +4 +2k/2log 10. +(28) +The inequality +4 +2k/2log 10 > +1 +2 · (n − 2)2 +implies that k < 2000. +Assume that +4 +2k/2log 10 ≤ +1 +2 · (n − 2)2 . +10 + +Then d1 − 1 +n − 2 is a convergent of +log 2 +log 10, say pi/qi. Then qi < n − 2 < +1.8 · 10291 implies i < 588 and max ai = 5393. So, from the properties of +continued fractions, see [16, Theorem 1.1.(iv)], +2k/2 < 4 · 5395 · 1.8 · 10291 +log 10 +< 1.7 · 10295 < 2981. +Thus, the upper bound k < 2000 is valid in this case also. +If (a, b, d1 − d2) = (9, 5, 1) then +τ + µ(9,5,1) := log 2 +log 10 − +log (1 + −4 +10 )(1/3) +log 10 += 1 ∈ Z, +and hence +Γ4 += |d1 log 10 − (n − 2) log 2 − log(2/10)| += |(d1 − 1) log 10 − (n − 3) log 2| . +If (a, b, d1 − d2) = (9, 5, 2), then +5τ + µ(9,5,1) := 5 log 2 +log 10 − +log (1 − +4 +100 )(1/3) +log 10 += 2 ∈ Z. +In this case, we write +Γ4 += |d1 log 10 − (n − 2) log 2 + log(32/100)| += |(d1 − 2) log 10 − (n − 7) log 2| , +and hence +0 < +���� +log 2 +log 10 − d1 − 2 +n − 7 +���� < +4 +2k/2log 10. +(29) +Similar to the first one, we see that the bound k < 2000 is also valid +in these two cases also. +We repeat the same reduction steps one more time but taking k < 2000 +and M := 8.5 · 1060 > n. When we work on (23), this time, we take q129 +instead of q588 and we find that +ǫa := ||µaq129|| − M||τq129|| > 0.031955, +for each a ∈ {1, 2, . . . , 9}. +By Lemma 3 we obtain λ < 210. +Hence, λ = k/2−5 means that k < 430. Assume that λ = (d1 − d2)log(10) +log(2) − +8. Then +d1 − d2 < 70. +Now, we pass to the Γ4, and we take q135 instead of q595. Then we find +that +ǫ(a,b,d1−d2) := ||µ(a,b,d1−d2)q135|| − M||τq135|| ≥ ǫ(4,6,2) > 0.000065 +for each a ∈ {1, 2, . . . , 9}, b ∈ {1, 2, . . . , 9}, d1 − d2 ∈ {1, 2, . . . , 70}, except +for the same three triples (a, b, d1 − d2) = (9, 2, 1), (9, 5, 1), (9, 5, 2). Thus, +we repeat the same calculations as we did before and we find that, even +in the exceptional cases, k < 460 which contradicts the fact k > 470. So, +we conclude that Equation (1) has no solutions when k > 470 and a ̸= 0. +11 + +4.6 +The Case a = 0 and k−Lucas Numbers of the +form b10d2 +Let a = 0. Then (1) turns into the equation +L(k) +n += b10d2. +(30) +Clearly, we take b > 0. In fact, our previous work contains most of +the material to solve this equation, with some small manipulation on the +variables. So, in any applicable case, we follow the previous notation to +prevent the recalculation. +By (30), Λ2 which was given in (11) is valid as +|Λ3 +2| := |α−(n−1)10d2fk(α)−1(2α − 1)−1b − 1| ≤ +5 +2αn−1 , +and Λ +′ +2 ̸= 0. Let +η1 := α, η2 := 10, η3 := fk(α)−1(2α − 1)−1b +with b1 := −(n − 1), b2 := d2, b3 := 1. So, +h(η3) ≤ h(fk(α)−1)+h((2α−1)−1)+h(b) ≤ 3 log k+log 3+log 9 < 8 log k. +From (30) and (6), we may write 10d2 ≤ L(k) +n +≤ 2αn < 2n. Thus, it is +enough to take B := n − 1. Note that, the inequalities 1 + log k < 3 log k +and 1 + log n − 1 < 2 log n − 1 holds for all k ≥ 2 and n ≥ 4. We apply +Theorem 2 by following the similar notation as we did before for Λ2, we +obtain that +n − 1 < 1.6 × 1013k4 log2 k log(n − 1). +We take T := 1.6 × 1013k4 log2 k. Then log T < 60 log k for all k ≥ 2. +Thus, from Lemma 4, we find +n < 2.1 × 1017k4 log4 k. +(31) +Assume that k ≤ 450, then n < 3 × 1029. By repeating the similar +calculations, as we did before for (15) to the inequality, +0 < +����(n − 1) log α +log 10 − d2 − log(bfk(α)−1(2α − 1)−1 +log 10 +���� < +5 +αn−1 log 10, +we see that the bounds found for a ̸= 0 strictly hold for the case a = 0. +Hence, by a computer search, we see that (30) has no solution when +k ≤ 470. +Let k > 450. From (16), we write +0 ̸= Λ′ +4 := +���2−(n−2)10d2b/3 − 1 +��� ≤ +1 +2k/2 . +By taking +(η1, |b1|) := (2, n − 2), (η2, |b2|) := (10, d2), (η3, |b3|) := (b/3, 1), +12 + +from Theorem 2 together with (31), we find k < 4 × 1014 and hence, from +(31), n < 6.9 × 1081. To reduce these bounds, we write +Γ′ +4 := |(n − 2) log 2 − d2 log 10 − log(b/3)| , +so that, as we did before, we obtain +0 < +����(n − 2) log 2 +log 10 − d2 − log(b/3) +log 10 +���� < +2 +2k/2 log 10. +(32) +Assume that b ̸∈ {3, 6}. +Then, applying Lemma 3 by choosing the parameters as M := 6.9 × +1081, µb := − log(b/3)/ log 10, ǫb := ||µbq170|| − M||τq170|| and the others +as in the previous section, we find that k < 564. If b is 3 or 6 then, from +Γ′ +4, we have that +���� +log 2 +log 10 − u +v +���� < +2 +2k/2v log 10, +where u +v is +d2 +n − 2 and +d2 +n − 3, respectively. We use the theory of continued +fractions as we did before for (28), to obtain that k < 572. Thus, from +(31), we obtain a reduced bound as n < 4 × 1031. We repeat the same +reduction algorithm with M := 4 × 1031 and as a result we obtain that +k < 440, a contradiction. This completes the proof. +References +[1] Alahmadi, A.; Altassan, A.; Luca, F.; Shoaib, H. k-generalized Fi- +bonacci numbers which are concatenations of two repdigits. Glasnik +matematiˇcki 2021, 56, 29-46. +[2] Altassan, A., Alan, M. Almost Repdigit k-Fibonacci Numbers with +an Application of k-Generalized Fibonacci Sequences, Mathematics, +(2023), 11(2), 455 +[3] Baker, A.; Davenport, H. The equations 3x2 −2 = y2 and 8x2 −7 = z2 +Quart. J. Math. Oxford Ser. 1969, 20, 129-137. +[4] Bednaˇr´ık, D.; Trojovsk´a, E. Repdigits as product of Fibonacci and +Tribonacci numbers. Mathematics, 2020, 8, 1720. +[5] Bravo, E.F.; Bravo, J.J.; G´omez, C. A. Generalized Lucas Numbers +Which are Concatenations of Two Repdigits. Results in Mathematics +2021 76, 1-16. +[6] Bravo, J.J., G´omez, C.A.G., Luca, F.: Powers of two as sums of two +k-Fibonacci numbers. Miskolc Math. Notes 17, 85–100 (2016) +[7] Bravo, J.J., G´omez, C.A., Luca, F.: A Diophantine equation in k- +Fibonacci numbers and repdigits. 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Fibonacci +Quart. 36(2), 129–145 (1998) +14 + diff --git a/ktE9T4oBgHgl3EQfVheR/content/tmp_files/load_file.txt b/ktE9T4oBgHgl3EQfVheR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..01da0f3b198c745a6db261ebfae5cd923d773511 --- /dev/null +++ b/ktE9T4oBgHgl3EQfVheR/content/tmp_files/load_file.txt @@ -0,0 +1,490 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf,len=489 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='07334v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='NT] 18 Jan 2023 Almost Repdigits in k−generalized Lucas Sequences Alaa ALTASSAN and Murat ALAN King Abdulaziz University Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Jeddah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 21589,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Saudi Arabia Yildiz Technical University Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Istanbul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 34210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Turkey January 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 2023 Abstract Let k ≥ 2 and (L(k) n )n≥2−k be the k−generalized Lucas sequence with initial condition L(k) 2−k = · · · = L(k) −1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' L(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 0 = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' L(k) 1 = 1 and each term afterwards is the sum of the k preceding terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' A positive integer is an almost repdigit if its digits are all equal except for at most one digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In this paper, we work on the problem of determining all terms of k−generalized Lucas sequences which are almost repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In particular, we find all k−generalized Lucas numbers which are powers of 10 as a special case of almost repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Key Words: k−Lucas numbers, repdigits, almost repdigits, linear forms in logarithms 2010 Mathematics Subject Classification: 11B39, 11J86, 11D61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 1 Introduction Let k ≥ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' The k−generalized Lucas sequence or, for simplic- ity, the k−Lucas sequence is a sequence given by the recurrence relation L(k) n = L(k) n−1 + · · · + L(k) n−k for all n ≥ 2, with the initial values L(k) i = 0 for i = 2 − k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , −1, L(k) 0 = 2 and L(k) 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We call the terms of this sequence k−Lucas numbers for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' For k = 2, this sequence is the classical Lucas sequence, and hence the k−Lucas sequence is a generalization of the Lucas sequence from binary recurrence sequence to the order k recurrence sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Recall that, a positive integer whose all digits are equal is called a repdigit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In recent years, many researches have been performed to find all terms of some sequences related to repdigits, see for example [1, 4, 5, 9, 10, 11, 12, 19, 20, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In this study, we search the numbers similar to the repdigits in k−Lucas numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 1 In [14] and [18], all square and perfect power positive integers whose digits are all equal except for one digit have been examined without giving a specific name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We call a positive integer an almost repdigit if its all digits are equal except for at most one digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' This numbers can be written of the form a �10d1 − 1 9 � + (b − a)10d2, 0 ≤ d2 < d1 and 0 ≤ a, b ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Usual repdigits and the numbers of the form b10d2 are two particular cases of almost repdigits which corresponds to the cases a = 0 and b = a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, almost repdigits are a generalization of repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Recently, in [2], the authors found all k−generalized Fibonacci num- bers that are almost repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In this paper, we continue to search almost repdigits by taking into account k−generalized Lucas numbers as an ana- logue of the study in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In other words, we consider the Diophantine equation L(k) n = a �10d1 − 1 9 � +(b−a)10d2, 0 ≤ d2 < d1 and 0 ≤ a, b ≤ 9 (1) in non negative integers d1, d2, a and b and we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Theorem 1 The Diophantine equation (1) has solutions only in the cases L(2) 11 = 199, L(2) 12 = 322, L(3) 8 = 118, L(3) 10 = 399, L(7) 10 = 755 and L(9) 10 = 766, when L(k) n has at least three digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Since, the numbers having at most two digits are trivially almost repdigits, we state the above theorem only for L(k) n which are consist of at least three digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, from now on we take d1 ≥ 3 and hence n > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' The proof of the above theorem, mainly depends on two effective meth- ods, that is, the linear forms in logarithms of algebraic numbers due to Matveev [21] as well as reduction algorithm due to Dujella and Peth˝o [13], which is in fact originally introduced by Baker and Davenport in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We give some details of these methods in the next section whereas in the third section, we give the main properties of k−Lucas sequences that we will need later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We devoted the forth section to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' It is also worth to note that, we implemented the software Maple for all calculations and computations in the proof the Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 2 The Tools Let θ be an algebraic number, and let c0xd + c1xd−1 + · · · + cd = c0 d � i=1 (x − θ(i)) be its minimal polynomial over Z, with degree d, where the ci’s are rela- tively prime integers with c0 > 0, and the θ(i)’s are conjugates of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 2 The logarithmic height of θ is defined by h(θ) = 1 d � log c0 + d � i=1 log � max{|θ(i)|, 1} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If θ = r/s is a rational number with relatively prime integers r and s and s > 0, then h(r/s) = log max{|r|, s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' The following properties are very useful in calculation of a logarithmic height : h(θ1 ± θ2) ≤ h(θ1) + h(θ2) + log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' h(θ1θ±1 2 ) ≤ h(θ1) + h(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' h(θs) = |s|h(θ), s ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Theorem 2 (Matveev’s Theorem) Assume that α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , αt are posi- tive real algebraic numbers in a real algebraic number field K of degree dK and let b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , bt be rational integers, such that Λ := αb1 1 · · · αbt t − 1, is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then |Λ| > exp � K(t)d2 K(1 + log dK)(1 + log B)A1 · · · At � , where K(t) := −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='4 · 30t+3 · t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='5 and B ≥ max{|b1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , |bt|}, and Ai ≥ max{dKh(αi), |log αi|, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='16}, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' For a real number θ, we put ||θ|| = min{|θ − n| : n ∈ Z}, which represents the distance from θ to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Now, we cite the following lemma which we will use it to reduce some upper bounds on the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Lemma 3 [6, Lemma 1] Let M be a positive integer, and let p/q be a convergent of the continued fraction of the irrational γ such that q > 6M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let A, B, µ be some real numbers with A > 0 and B > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If ǫ := ||µq|| − M||γq|| > 0, then there is no solution to the inequality 0 < |uγ − v + µ| < AB−w, in positive integers u, v and w with u ≤ M and w ≥ log(Aq/ǫ) log B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We cite the following lemma from [15, Lemma 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Lemma 4 Let m ≥ 1 and T > (4m2)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then we have x (log x)m < T ⇒ x < 2mT (log(T ))m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 3 3 Properties of k−Fibonacci and k−Lucas Numbers The characteristic polynomial of k−Lucas numbers is Ψk(x) = xk − xk−1 − · · · − x − 1 which is an irreducible polynomial over Q[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' The polynomial Ψk(x) has exactly one real distinguished root α(k) outside the unit circle [22, 23, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' The other roots of Ψk(x) are strictly inside the unit circle [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' This root α(k), say α for simplicity, placed in the interval 2(1 − 2−k) < α < 2 for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let fk(x) = x − 1 2 + (k + 1)(x − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (2) It is known that the inequalities 1/2 < fk(α) < 3/4 and |fk(αi)| < 1, 2 ≤ i ≤ k (3) are hold, where α := α1, · · · , αk are all the roots of Ψk(x) [6, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In particular, we deduce that fk(α) is not an algebraic integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In the same Lemma, it is also proved that h(fk(α)) < 3 log(k) holds ∀k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (4) In [10], Bravo and Luca showed that L(k) n = k � i=1 (2αi − 1)fk(α)(αi)n−1 and ���L(k) n − (2α − 1)fk(α) ��� < 3/2 (5) for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' As in the classical k = 2 case, we have the similar bounds as αn−1 ≤ L(k) n ≤ 2αn (6) for all n ≥ 1 and k ≥ 2 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 4 Proof of Theorem 1 Assume that Equation (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By combining the inequality 10d1−2 < a �10d1 − 1 9 � + (b − a)10d2 ≤ 2 · 10d1, and (6), we get that d1 < log 2 log 10(n + 1) + 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='31n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='31 < n − 1 (7) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='2n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='6 < log((1 + � (5))/2) log 10 (n − 1) − log 2 log 10 < d1 (8) for all n > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' First, we assume that a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We examine the case a = 0 in the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='1 The Case n < k + 1 and Almost Repdigits of the Form 3 · 2n Assume that n ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In this case, L(k) n = 3 · 2n−2, and hence Equation (1) can be written as 27 · 2n−2 = a � 10d1 − 1 � + 9(b − a)10d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Taking modulo 2d2 and modulo 2d1, we find that d2 ≤ 3 and d1 ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Hence, from (8), we see that n < 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' A quick calculation shows that when n < 70, there is no almost repdigits of the form 3 · 2n−2 with at least three digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, from now on, we take n ≥ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='2 A Bound for n Depending on k By rewriting (1) as L(k) n + a/9 − (b − a)10d2 = a10d1/9, and by using (5), we get ���fk(α)(2α − 1)αn−1 − a10d1/9 ��� ≤ (3/2) + ���a/9 − (b − a)10d2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Therefore, we obtain |Λ1| ≤ 27/2 10d1 + 1 10d1 + |b − a|(9/a) 10d1−d2 ≤ 87 10d1−d2 , (9) where Λ1 := αn−110−d1fk(α)(2α − 1)9/a − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We take η1 := α, η2 := 10, η3 := fk(α)(2α − 1)9/a, b1 := n − 1, b2 := −d1, b3 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Note that from (4), we find h(η3) ≤ h(9/a) + h(fk(α)) + h(2α − 1) < 8 log(k), since h(2α − 1) < log 3 [10, page 147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We have also Λ1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Indeed, if Λ1 = 0, then we would get a10d1/9 = fk(α)(2α − 1)αn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Conjugating both sides of this relation by any one of the automorphisms σi : α → αi for any i ≥ 2 and by taking the absolute values, we find that 100 < |fk(αi)||2α − 1||αi|n−1 < 3, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus,Λ1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Other calculations are doneby using similar techniques as in the k−Fibonacci case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, by combining the result of Theorem 2 and the fact that log (Λ1) < log 87−(d1 − d2) log 10, we obtain d1 − d2 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='8 · 1012 · k4(log2 k) log (n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (10) 5 By rearranging Equation (1) as follows L(k) n + a/9 = a10d1/9 + (b − a)10d2, and using (5), we get |Λ2| ≤ 5 2 1 fk(α)αn−1 ≤ 5 2αn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (11) where Λ2 := α−(n−1)10d1fk(α)−1(2α − 1)−1((a/9) + (b − a)10d2−d1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By the similar argument as above we see that Λ2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let η1 := α, η2 := 10, η3 := fk(α)−1(2α − 1)−1((a/9) + (b − a)10d2−d1) with b1 := −(n − 1), b2 := d1, b3 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' All η1, η2 and η3 belong to the real number field K = Q(α) and therefore we take dK = 2, to be the degree of the number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Using the properties of logarithmic height, and the fact that h(2α − 1) < log 3, we find h(η3) ≤ h(fk(α)−1) + h((2α − 1)−1) + h((a/9) + (b − a)10d2−d1) ≤ 3 log(k) + h((2α − 1)) + h(a/9) + h(b − a) + h(10d2−d1) + log(2) ≤ 3 log(k) + log 3 + log(144) + |d2 − d1| log(10) < 12 log(k) + |d2 − d1| log(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By Theorem 2, we get a bound for log(Λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then by combining this bound with the one comes from (11), we get n − 1 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='4 · 1025k8(log(k))3(log(n − 1))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Therefore, from Lemma 4, we have n < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='3 · 1030k8(log(k))5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (12) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='3 The Case k ≤ 470 Let Γ1 := (n − 1) log α − d1 log 10 + log(fk(α)(2α − 1) · 9/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then Λ1 := |exp(Γ1) − 1| < 87/10d1−d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We claim that d1 − d2 < 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Suppose that d1 − d2 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then 87/10d1−d2 < 1/2 and therefore |Γ1| < 174/10d1−d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So we have 0 < ����(n − 1) log α log 10 − d1 + log(fk(α) · (2α − 1) · 9/a) log 10 ���� < 174/10d1−d2 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (13) For each 2 ≤ k ≤ 470, we take Mk := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='3 · 1030k8(log(k))5 > n and τk = log α log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then, for each k, we find a convergent pi/qi of the continued fraction of irrational τk such that qi > 6Mk After that, we calculate ǫ(k,a) := ||µ(k,a)qi|| − Mk||τkqi|| for each a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}, where µ(k,a) := log(fk(α) · (2α − 1) · 9/a) log 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 6 If ǫ(k,a) < 0, then we repeat the same calculation for qi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Except for (k, a) = (2, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In fact 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='00008 < ǫ(k,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, from Lemma 3, we find an upper bound on d1 − d2 for each 2 ≤ k ≤ 470 such that none of them are greater than 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, we conclude that d1 − d2 < 65 as we claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If (k, a) = (2, 9), then τ = µ(k,a) and hence ǫ(k,a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, in this case, (13) is 0 < ����n log α log 10 − d1 ���� < 174/10d1−d2 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In other words, 0 < ���� log α log 10 − d1 n ���� < 174/10d1−d2 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (14) From, the theory of continued fractions, we see that this implies that the bound d1 − d2 < 65 is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Next, let Γ2 := −(n−1) log α+d1 log 10+log(fk(α)−1·(2α−1)−1·((a/9)+(b−a)10d2−d1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, Λ2 := |exp(Γ2) − 1| < 5/2αn−1 < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Hence, we get that 0 < ����(n − 1) log α log 10 − d1 + µ(k,d1−d2,a,b) ���� < 5 αn−1 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (15) where µ(k,d1−d2,a,b) := −log(fk(α)−1 · (2α − 1)−1 · ((a/9) + (b − a)10d2−d1)) log 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' This time we calculate ǫ(k,d1−d2,a,b) := ||µ(k,d1−d2,a,b)qi|| − Mk||τkqi|| for each d1 − d2 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 65}, a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9} and b ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We apply Lemma 3 to Equation (15), and therefore we find an upper bound on n−1 for each 2 ≤ k ≤ 470, say nL(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' For example nL(3) < 150, nL(10) < 147, nL(100) < 178, nL(200) < 197, nL(300) < 296, nL(400) < 396 and n(470) < 465 are some of these bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By writing a short computer programme in Maple, and using the obtained bounds, we find that L(2) 11 = 199, L(2) 12 = 322, L(3) 8 = 118, L(3) 10 = 399, L(7) 10 = 755 and L(9) 10 = 766, are the only k−Lucas num- bers which are almost repdigits with at least three digits, as we claimed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Now, we turn our focus to the case k > 470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='4 The Case k > 470 We use the following lemma [24, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Lemma 5 If n < 2k/2, then the following estimates hold: L(k) n = 3 · 2n−2(1 + ζ(n, k)), where |ζ(n, k)| < 1 2k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 7 For k > 470, the inequality n < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='3 · 1030k8(log(k))5 < 2k/2, holds and hence from the above Lemma, we have that ���3 · 2n−2 − L(k) n ��� < 3 · 2n−2 2k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (16) Now, we turn back to (1) one more time to rewrite it as ���L(k) n − (a/9)10d1 ��� < (a/9) + |b − a|10d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (17) Thus, by combining (16) and (17), we get ���3 · 2n−2 − (a/9)10d1 ��� < 3 · 2n−2 2k/2 + (a/9) + |b − a|10d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Therefore, we have Λ3 := ���2n−210−d127/a − 1 ��� < 1 2λ , (18) where λ := min{(k/2) − 5, (d1 − d2)log(10) log(2) − 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let η1 := 2, η2 := 10, η3 := 27/a and b1 := n − 2, b2 := −d1, b3 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Applying Theorem 2 to Λ3, we get λ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='2 · 1012 log n where we used log n < log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='3 · 1030k8(log(k))5) < log(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='3) + 3 log(10) + 8 log k + 5 log log(k) < 50 log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, if λ := (k/2) − 5, then we get a bound for k as k < 1016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If λ := (d1 − d2)log(10) log(2) − 8, then we get d1 − d2 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='4 · 1013 log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (19) This bound of d1 − d2 also leads to an upper bound of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' To do this, we rewrite (1) as ���L(k) n − (a/9)10d1 − (b − a)10d2 ��� ≤ (a/9) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (20) By combining (20) with (16), we find that Λ4 := ���2−(n−2)10d1((a/9) + (b − a)10d2−d1)(1/3) − 1 ��� < 1 3 · 2n−2 2 2k/2 ≤ 2 2k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We take η1 := 2, η2 := 10, η3 := (1/3)((a/9) + (b − a)10d2−d1) 8 and b1 := −(n − 2), b2 := −d1, b3 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then h(η3) = h(a/9) + h(b − a) + |d2 − d1|h(10) + h(3) + log 2 < log 432 + (d1 − d2) log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Other calculations are similar to those for Λ4 as K = Q, dK = 1, B := n > n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' h(η1) = log 2, h(η2) = log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Moreover Λ4 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Indeed, 2n−2 = 10d1(a/9) + (b − a)10d2 implies that a = 9 and d2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' For d1 = 3, clearly the equation 2n−2 = 10d1 + b − 9 has no solution in integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So d1 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, congruence consideration modulo 24 shows that this equation has no integer solutions for 0 ≤ b ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Hence, Λ4 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Therefore, Theorem 2 together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='4) give that log 2−(k/2) log 2 > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='4·306·34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='5(1+log n) log 2·log 10·(log 432+(d1 − d2) log 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' At this point, we use the upper bound of d1 − d2 which was given in (19), and by using the estimates log 432 < log k and log n < 50 log k, we obtain the desired upper bound for k as k < 3 · 1031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (21) Thus, by (12), we have also a bound for n as n < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='8 · 10291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (22) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='5 Reducing the Bound on k We will reduce these highly large upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let Γ3 := (n − 2) log 2 − d1 log 10 + log(27/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (23) Then Λ3 := |exp(Γ3) − 1| < 1 2λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We will find a feasible bound for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Suppose that λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then, 1 2λ < 1 2 and hence we get that |Γ3| < 2 2λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In this case, we don’t need to consider the case a = 9 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' From (23), we write 0 < ����(n − 2) log 2 log 10 − d1 + log(27/a) log 10 ���� < 2 2λ log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (24) Let M := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='8 · 10291 > n and τ = log 2 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then, the denominator of the 588th convergent of τ, say q588, exceeds 6M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then ǫa := ||µaq588|| − M||τq588|| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='029559, for each a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}, where µa := log(27/a)/log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, by applying Lemma 3, we get λ < 975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 9 Hence, if λ = k/2−5, then k < 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Assume that λ = (d1 − d2)log(10) log(2) − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then d1 − d2 < 296 < 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let Γ4 = ���(n − 2) log 2 − d1 log 10 − log((a/9) + (b − a)10d2−d1)(1/3) ��� (25) Then Λ4 := |exp(Γ4) − 1| < 2 2k/2 < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (26) So 0 < | Γ4 log 10| < ����(n − 2) log 2 log 10 − d1 + µ(a,b,d1−d2) ���� < 4 2k/2log 10 (27) where µ(a,b,d1−d2) := −log ((a/9) + (b − a)10d2−d1)(1/3) log 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We take M := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='8 · 10291 > n and τ = log 2 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' This time, we take q595, which is the denominator of the 595th con- vergent of τ, as qi > 6M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let ǫ(a,b,d1−d2) := ||µ(a,b,d1−d2)q595||−M||τq595|| for each a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}, b ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9} and d1 − d2 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 300}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We find that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='000036 < ǫ(6,0,272) ≤ ǫ(a,b,d1−d2) for all a, b, d1 − d2 except for (a, b, d1 − d2) = (9, 2, 1), (9, 5, 1), (9, 5, 2) since these three triples of (a, b, d1 − d2), we have that µ(a,b,d1−d2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then, except for the above three triples, from Lemma 3, we conclude that k < 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Hence, from (12), n < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='5 · 1060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If (a, b, d1 − d2) = (9, 2, 1), then µ(9,2,1) := − log (1 − 7 10)(1/3) log 10 = 1 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, in this case we may write Γ4 = ��(n − 2) log 2 − d1 log 10 − log 10−1�� = |(n − 2) log 2 − (d1 − 1) log 10| , and hence 0 < ���� log 2 log 10 − d1 − 1 n − 2 ���� < 4 2k/2log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (28) The inequality 4 2k/2log 10 > 1 2 · (n − 2)2 implies that k < 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Assume that 4 2k/2log 10 ≤ 1 2 · (n − 2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 10 Then d1 − 1 n − 2 is a convergent of log 2 log 10, say pi/qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then qi < n − 2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='8 · 10291 implies i < 588 and max ai = 5393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, from the properties of continued fractions, see [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (iv)], 2k/2 < 4 · 5395 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='8 · 10291 log 10 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='7 · 10295 < 2981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, the upper bound k < 2000 is valid in this case also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If (a, b, d1 − d2) = (9, 5, 1) then τ + µ(9,5,1) := log 2 log 10 − log (1 + −4 10 )(1/3) log 10 = 1 ∈ Z, and hence Γ4 = |d1 log 10 − (n − 2) log 2 − log(2/10)| = |(d1 − 1) log 10 − (n − 3) log 2| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If (a, b, d1 − d2) = (9, 5, 2), then 5τ + µ(9,5,1) := 5 log 2 log 10 − log (1 − 4 100 )(1/3) log 10 = 2 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In this case, we write Γ4 = |d1 log 10 − (n − 2) log 2 + log(32/100)| = |(d1 − 2) log 10 − (n − 7) log 2| , and hence 0 < ���� log 2 log 10 − d1 − 2 n − 7 ���� < 4 2k/2log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (29) Similar to the first one, we see that the bound k < 2000 is also valid in these two cases also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We repeat the same reduction steps one more time but taking k < 2000 and M := 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='5 · 1060 > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' When we work on (23), this time, we take q129 instead of q588 and we find that ǫa := ||µaq129|| − M||τq129|| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='031955, for each a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By Lemma 3 we obtain λ < 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Hence, λ = k/2−5 means that k < 430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Assume that λ = (d1 − d2)log(10) log(2) − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then d1 − d2 < 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Now, we pass to the Γ4, and we take q135 instead of q595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then we find that ǫ(a,b,d1−d2) := ||µ(a,b,d1−d2)q135|| − M||τq135|| ≥ ǫ(4,6,2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='000065 for each a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}, b ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 9}, d1 − d2 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' , 70}, except for the same three triples (a, b, d1 − d2) = (9, 2, 1), (9, 5, 1), (9, 5, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, we repeat the same calculations as we did before and we find that, even in the exceptional cases, k < 460 which contradicts the fact k > 470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, we conclude that Equation (1) has no solutions when k > 470 and a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='6 The Case a = 0 and k−Lucas Numbers of the form b10d2 Let a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then (1) turns into the equation L(k) n = b10d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (30) Clearly, we take b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' In fact, our previous work contains most of the material to solve this equation, with some small manipulation on the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, in any applicable case, we follow the previous notation to prevent the recalculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By (30), Λ2 which was given in (11) is valid as |Λ3 2| := |α−(n−1)10d2fk(α)−1(2α − 1)−1b − 1| ≤ 5 2αn−1 , and Λ ′ 2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let η1 := α, η2 := 10, η3 := fk(α)−1(2α − 1)−1b with b1 := −(n − 1), b2 := d2, b3 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' So, h(η3) ≤ h(fk(α)−1)+h((2α−1)−1)+h(b) ≤ 3 log k+log 3+log 9 < 8 log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' From (30) and (6), we may write 10d2 ≤ L(k) n ≤ 2αn < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, it is enough to take B := n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Note that, the inequalities 1 + log k < 3 log k and 1 + log n − 1 < 2 log n − 1 holds for all k ≥ 2 and n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We apply Theorem 2 by following the similar notation as we did before for Λ2, we obtain that n − 1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='6 × 1013k4 log2 k log(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We take T := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='6 × 1013k4 log2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then log T < 60 log k for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, from Lemma 4, we find n < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='1 × 1017k4 log4 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (31) Assume that k ≤ 450, then n < 3 × 1029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By repeating the similar calculations, as we did before for (15) to the inequality, 0 < ����(n − 1) log α log 10 − d2 − log(bfk(α)−1(2α − 1)−1 log 10 ���� < 5 αn−1 log 10, we see that the bounds found for a ̸= 0 strictly hold for the case a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Hence, by a computer search, we see that (30) has no solution when k ≤ 470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Let k > 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' From (16), we write 0 ̸= Λ′ 4 := ���2−(n−2)10d2b/3 − 1 ��� ≤ 1 2k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' By taking (η1, |b1|) := (2, n − 2), (η2, |b2|) := (10, d2), (η3, |b3|) := (b/3, 1), 12 from Theorem 2 together with (31), we find k < 4 × 1014 and hence, from (31), n < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='9 × 1081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' To reduce these bounds, we write Γ′ 4 := |(n − 2) log 2 − d2 log 10 − log(b/3)| , so that, as we did before, we obtain 0 < ����(n − 2) log 2 log 10 − d2 − log(b/3) log 10 ���� < 2 2k/2 log 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' (32) Assume that b ̸∈ {3, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Then, applying Lemma 3 by choosing the parameters as M := 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='9 × 1081, µb := − log(b/3)/ log 10, ǫb := ||µbq170|| − M||τq170|| and the others as in the previous section, we find that k < 564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' If b is 3 or 6 then, from Γ′ 4, we have that ���� log 2 log 10 − u v ���� < 2 2k/2v log 10, where u v is d2 n − 2 and d2 n − 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We use the theory of continued fractions as we did before for (28), to obtain that k < 572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Thus, from (31), we obtain a reduced bound as n < 4 × 1031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' We repeat the same reduction algorithm with M := 4 × 1031 and as a result we obtain that k < 440, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' References [1] Alahmadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Altassan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Luca, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Shoaib, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' k-generalized Fi- bonacci numbers which are concatenations of two repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Glasnik matematiˇcki 2021, 56, 29-46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [2] Altassan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=', Alan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Almost Repdigit k-Fibonacci Numbers with an Application of k-Generalized Fibonacci Sequences, Mathematics, (2023), 11(2), 455 [3] Baker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Davenport, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' The equations 3x2 −2 = y2 and 8x2 −7 = z2 Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Oxford Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 1969, 20, 129-137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [4] Bednaˇr´ık, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Trojovsk´a, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Repdigits as product of Fibonacci and Tribonacci numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Mathematics, 2020, 8, 1720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [5] Bravo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Bravo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' G´omez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Generalized Lucas Numbers Which are Concatenations of Two Repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Results in Mathematics 2021 76, 1-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [6] Bravo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=', G´omez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 2022, 84, 125-145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [25] Rihane, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' k-Fibonacci and k-Lucas Numbers as Product of Two Repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 2021, 76), 1-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [26] S¸iar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Keskin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' k-Generalized Pell Numbers Which are Concate- nation of Two Repdigits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Mediterranean Journal of Mathematics 2022, 19, 1-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' [27] Wolfram, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' : Solving generalized Fibonacci recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' Fibonacci Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} +page_content=' 36(2), 129–145 (1998) 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE9T4oBgHgl3EQfVheR/content/2301.07334v1.pdf'} diff --git a/l9AyT4oBgHgl3EQfyfn1/content/tmp_files/2301.00687v1.pdf.txt b/l9AyT4oBgHgl3EQfyfn1/content/tmp_files/2301.00687v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a66839cd55b46758bec6ab8891a559c266bd023 --- /dev/null +++ b/l9AyT4oBgHgl3EQfyfn1/content/tmp_files/2301.00687v1.pdf.txt @@ -0,0 +1,922 @@ +arXiv:2301.00687v1 [hep-th] 30 Dec 2022 +Quantum Thermodynamics of an α′-Corrected +Reissner-Nordstr¨om Black Hole +Behnam Pourhassan,a ˙Izzet Sakallı,b Xiaoping Shi,c Mir Faizal,d Salman Sajad Wanie +aSchool of Physics, Damghan University, Damghan, 3671641167, Iran. +bPhysics Department, Eastern Mediterranean University, Famagusta 99628, North Cyprus via Mersin 10, +Turkey. +c,dIrving K. Barber School of Arts and Sciences, University of British Columbia Okanagan, Kelowna, BC +V1V 1V7, Canada. +dCanadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada. +eDepartment of Physics Engineering, Istanbul Technical University, Istanbul, 34469 Turkey. +E-mail: b.pourhassan@du.ac.ir, izzet.sakalli@emu.edu.tr, +mirfaizalmir@gmail.com +Abstract: In this paper, we will analyze the effects of α′ corrections on the behavior of a Reissner- +Nordstr¨om black hole. We will calculate the effects of such corrections on the thermodynamics +and thermodynamic stability of such a black hole. We will also derived a novel α′-corrected first +law. +We will investigate the effect of such corrections on the Parikh-Wilczek formalism. +This +will be done using cross entropy and Kullback-Leibler divergence between the original probability +distribution and the α′-corrected probability distribution. We will then analyze the non-equilibrium +quantum thermodynamics of this black hole. It will be observed that its quantum thermodynamics +is corrected due to quantum gravitational corrections. We will use Ramsey scheme for emitted +particles to calculate the quantum work distribution for this system. The average quantum work +will be related to the difference of α′-corrected free energies using the Jarzynski equality. + +Contents +1 +Introduction +1 +2 +Quantum Gravitational Corrections +3 +3 +Evaporation +6 +4 +Quantum Work Distribution +9 +5 +Conclusion +11 +1 +Introduction +The equilibrium thermodynamics of black holes is well suited to describe the behavior of black holes +at large distances [1–4]. In this equilibrium description, temperature of the Hawking radiation is +inversely proportional to its surface gravity, and the entropy of the black hole scales with its area. +So, as the size of the black hole reduces, its temperature increases. At such short distances, due +to high temperature, the effect of thermal fluctuations cannot be neglected. Due to these thermal +fluctuations the equilibrium description breaks down [5–10]. It is known that non-equilibrium quan- +tum thermodynamics can be used to describe a systems where the equilibrium description breaks +down [11–14]. This has motivated the use of non-equilibrium quantum thermodynamics to describe +the behavior of small black holes [15, 16]. In the original Bekenstein-Hawking entropy, a semi- +classical approximation is used and the quantum gravitational corrections are neglected [17, 18]. +Even though this is well suited for large distance physics of black holes, it is important to consider +quantum gravitationally corrected entropy at short distances [19–21]. As the entropy of black holes +is related to the holographic principle [22, 23], these quantum gravitational corrections can also be +holographically obtained using the AdS/CFT correspondence [24–30]. The quantum gravitation- +ally corrected thermodynamics for small scales black holes has been studied using string theoretical +effects [31–33]. It is also possible to use properties of conformal field theory to obtain quantum +gravitational corrections to black hole entropy at short distances [34–37]. The consequences of such +quantum gravitational corrections on the behavior of black branes has also been studied [38, 39]. It +was observed that at large distances, these quantum gravitational corrections for black branes do +not change their thermodynamics, and their effects can be neglected. However, their effects at short +distances cannot be neglected, and they produce non-trivial modifications to the thermodynamics +of black branes. This is similar to the behavior of black holes [24–30]. +The non-equilibrium quantum thermodynamics of black solutions modified by quantum gravi- +tational corrections has been studied for small AdS black holes [40], Myers-Perry black hole [41], and +a system of M2-M5 branes [42]. This quantum work distribution was expressed using the difference +of quantum gravitationally corrected equilibrium free energies using the Jarzynski equality [43, 44]. +The Jarzynski equality is related to the quantum Crooks fluctuation theorem in non-equilibrium +quantum thermodynamics [45, 46]. The loss of unitarity in equilibrium black hole thermodynamics +has lead to the information paradox [47, 48]. However, equilibrium description breaks down at short +distances, and we have to use non-equilibrium quantum thermodynamics. We observed that the +Hawking radiation in black holes breaks unitarity as it represents heat, and heat in quantum ther- +modynamics is in turn represented by a non-unitary quantum process [49, 50]. However, quantum +– 1 – + +work distribution is represented by a unitary quantum process [49, 50], so at least there is a uni- +tary quantum process associated with black hole evaporation. Furthermore, this unitary quantum +process becomes important at short distances. Thus, it is possible that quantum thermodynamics +could potentially resolve the black hole information paradox [40–42]. +At short distance it is important to consider quantum gravitational corrections, and α′ correc- +tions represent such corrections in string theory. The α′ corrections to a black hole solution have +been obtained from the heterotic superstring effective action [51]. These α′ corrected solutions +were used to investigate the effect of these corrections on their equilibrium entropy. The effect of +such corrections on the entropy of both BPS and non-BPS extremal dyonic black holes in heterotic +string theory has also been discussed [52]. The near-horizon geometry of such solutions was used +to calculate the entropy of these corrected black hole solutions. The α′ corrected T-duality trans- +formations have been used to analyze the invariance of entropy for corrected solutions [53]. In this +paper, we will analyze the effects of α′ corrections on non-equilibrium quantum thermodynamics +of a Reissner-Nordstr¨om black hole solution. The α′ corrections to a four dimensional Reissner- +Nordstr¨om have been explicitly calculated [51]. This was done using heterotic superstring effective +field theory compactified on T 6. In fact, using this α′ corrected metric, the effect of α′ corrections on +the equilibrium entropy and temperature were also investigated. However, here we will generalize +this result by analyzing the effect of α′ corrections on the evaporation of this Reissner-Nordstr¨om +black hole, and its quantum thermodynamics. +The black hole information paradox has also been addressed by considering the self-gravitation +effects of the emitted particles on the black hole geometry [54]. Using this Parikh-Wilczek formalism, +it has been argued that the Hawking radiation could be a unitary non-thermal process [54]. In fact, +using the Parikh-Wilczek formalism the black hole entropy has been related to the entropy of such +emitted particles [55]. It is important to analyze the effects of α′ on Parikh-Wilczek formalism, +as the α′ can change the behavior of black hole during its last stages. So, in this paper, we will +investigate the effect of such quantum gravitational corrections on the Parikh-Wilczek formalism. +We will use Kullback-Leibler divergence [56, 57] and cross entropy [58, 59] to to analyze how +the probability density of particles emitted during evaporation is effected by α′ corrections. The +Kullback-Leibler divergence measures how different two probability distributions are from each +other. Furthermore, the Kullback-Leibler divergence can be related to cross entropy and entropy +of probability distributions. We will use these relations to obtain an expression for the effect of α′ +on such probability distribution. We will also use the Ramsey scheme to calculate the quantum +work distribution of these emitted particles [49, 50]. Thus, we will related average quantum work +in quantum thermodynamics of black hole [40–42] to the quantum work distribution of emitted +particles in the Parikh-Wilczek formalism [54, 55]. +It may be noted that the Parikh-Wilczek formalism has been used to investigate the non- +thermal unitary radiation from a Reissner-Nordstr¨om black hole [60]. +The thermodynamics of +Reissner-Nordstr¨om black hole is interesting [61], as the quantum corrections to its equilibrium +thermodynamics can be studied using relativistic quantum geometry [62]. It has been demonstrated +that small thermal fluctuations to the equilibrium thermodynamics of a Reissner-Nordstr¨om black +hole can modify its equilibrium entropy [63]. It is possible to relate these thermal fluctuations +to quantum fluctuations [5–9] using the Jacobson formalism formalism [64]. As the geometry an +emergent structure in Jacobson formalism, it is possible to use it to relate the quantum fluctuation +in the geometry to the thermal fluctuations in the equilibrium thermodynamics [65]. This formalism +has been used to obtain quantum fluctuations of different black hole solutions [66–69]. Thus, the +Jacobson formalism [64, 65] can also be used to obtain the quantum corrections to the geometry +of a Reissner-Nordstr¨om black hole from thermal fluctuations to its equilibrium thermodynamics +[63]. +The effect of a zero point length in geometry of spacetime on the thermodynamics of a +Reissner-Nordstr¨om black hole have been studied using the generalized uncertainty principle [70]. +– 2 – + +The generalized uncertainty principle is a modification to the usual quantum mechanical uncertainty +principle [71], and can be used to study quantum gravitational corrections to the semi-classical black +hole entropy [72]. The generalized uncertainty principle has been motivated from string theoretical +corrections [73, 74]. Thus, it is important to analyze the effects of quantum gravitational corrections +on the thermodynamics of a Reissner-Nordstr¨om black hole. So, in this paper, we will explicitly +analyze such effects produced by the α′ corrections on such a Reissner-Nordstr¨om black hole [51]. +2 +Quantum Gravitational Corrections +It is possible to obtain the α′ corrections to a four-dimensional Reissner-Nordstr¨om black hole using +heterotic superstring effective field theory compactified on T 6 [51]. We will analyze the quantum +thermodynamics of such a four dimensional Reissner-Nordstr¨om black hole. Now this solution will +be characterized by its mass M and its electric charge. Furthermore, it will also contain an explicit +α′-corrected contribution. The metric for such a solution can be expressed as [51], +ds2 = N 2f(r)dt2 − dr2 +f(r) − r2(dθ2 + sin2 θdϕ2), +(2.1) +where +N 2 = 1 + α′ p2 +8r4 +f(r) = 1 − 2M +r ++ p2 +2r2 − α′ p2 +4r4 +� +1 − 3M +2r + 11p2 +40r2 +� +. +(2.2) +and p is a physical constant related to the black hole charges. By neglecting α′ corrections, we +obtain the usual uncorrected radius of the event horizon radius [75], +r± = M +� +1 ± +� +1 − 1 +2 +� p +M +�2 +� +. +(2.3) +Now, the α′-corrected radius of the event horizon is rh = r+ + α′M −2r′ ++, where r′ ++ is defined as +[51] +r′ ++ ≡ M + + +−13 +20 +�M +p +�2 ++ 8 +5 +�M +p +�4 +− +9 +80 − 21 +20 +� +M +p +�2 ++ 8 +5 +� +M +p +�4 +� +1 − 1 +2 +� p +M +�2 + + + , +(2.4) +In the case of single electric charge q it is given by p = +√ +2q [76]. So, we will consider this case of a +single charge and assume p = +√ +2q for rest of this paper. Here, assuming q/M ≪ 1 and α′ = aM 2, +and so we can write rh = 2M +aM((M 2/5q2)−(9/80)). This represents the quantum gravitationally +corrected radius of the horizon for this black hole. We can write the corrected temperature of four +dimensional Reissner-Nordstr¨om black hole as following [51], +T = +γ +2πM (1 + γ)2 + α′ +M 2 +(1 + 3γ) (1 − γ)2 +160πMγ (1 + γ)5 . +(2.5) +Here, for the simplicity we defined γ ≡ +� +1 − (q/M)2. In Fig. 1 we plot the corrected temperature +of a four dimensional Reissner-Nordstr¨om black hole as a function of M to analyze its behavior. +We observe that for a large black hole, the α′ corrections do not change the temperature of the +black hole. This is expected as these are quantum gravitational corrections, and should only become +important at short distances. We also observe that at intermediate length scales the system becomes +– 3 – + +Figure 1. Temperature in terms of M with α′ = aM 2. +very sensitive to value of α′. However, at very short distances, this sensitive seems on α′ tends to +reduce. +The area of the event is Ah = 4πr2 +h, while the volume of is V = 4πr3 +h/3.. As the entropy of the +black hole is related to its area, we can express the entropy of this black hole as [51] +S = πM 2 +� +(1 + γ)2 + α′ +M 2 +21γ2 + 18γ + 1 +40γ (1 + γ) +� +. +(2.6) +To analyze the effects of quantum gravitational α′ corrections on the entropy of this system, we +define a µ-coefficient as the ratio of the corrected and original entropies, µS = S(α′)/S(α′ = 0). +This µ-coefficient measure the change in entropy due to α′ corrections +µS = 1 + α′ +M 2 +21γ2 + 18γ + 1 +40γ (1 + γ)3 +. +(2.7) +In Fig. 2, we have plotted µS for various values of the correction parameter (α′). We observe that +the effect of α′ corrections reduces as the size of the black hole increases. This is expected as these +corrections are quantum gravitational corrections, there effects can be neglected for large black +holes. We also see that the effect produced by α′ corrections seems to converge for small values +of mass. This seems to indicate that the effects gravitational corrections will become important +at small scales, and different values of α′ could lead to similar behavior. We now observe that +string tension can be taken as a variable string theory. In fact, the string tension has been studied +in various limits [77–80]. It has also been suggested that by a dynamical mechanism can be used +to obtain the value of the α′ [81, 82]. It has also been proposed that a bound on the α′ can be +obtained from high precision experiments involving low energy quantum mechanics systems [83– +86]. Thus, it is possible to view α′ as a variable and analyze its effects on thermodynamics. In +usual thermodynamics, different state variable their conjugate are used to analyze a thermodynamic +– 4 – + +Figure 2. µS in terms of M with q = 1. +system. This has been done for S, T and Φ, q. In fact, the negative cosmological constant along +with its thermodynamic conjugate has also been used as a state variable [87, 88]. Now if we take +the α′ as a variable, then to specify a state in such a system, we have to specify the value of the +α′ along with other thermodynamic variables. Thus, we can view the α′ as a thermodynamic state +variable, and define A as the conjugate of the α′, such that +� ∂S +∂M +� +q,α′ = 1 +T , +�∂S +∂q +� +M,α′ = −Φ +T , +� ∂S +∂α′ +� +M,q += −A +T . +(2.8) +As the entropy is depend on the parameters M, q and α′, we can write a novel α′-corrected first +law of thermodynamics as +dM = T dS + Φdq + Adα′. +(2.9) +This is the modification to the first law from quantum gravitational corrections. We obtain the +value of the corrected electrostatic potential conjugate to the black hole charge Φ as +Φ = +q +M (1 + γ) − α′q3 +80 +8M 4 (1 + γ) − M 2q2 (11 + 7γ) + 3q4 +M 9γ2 (1 + γ)7 +(2.10) +Similarly, by using the α′ as a state variable, we can obtain an expression for its conjugate A. Thus, +we can express A as +A = −160M 6 (1 + γ) − 80q2M 4 (5 + γ) + q4M 2 (321 + 181γ) − 3q6 (27 + 7γ) +80M 7γ2 (1 + γ)6 +− α′ � +16M 4 (γ − 1) + 4q2M 2 (17 − γ) + q4 (63γ − 51) +� +6400M 7γ2 (1 + γ)6 +(2.11) +– 5 – + +Now we can write a quantum gravitationally corrected Smarr-Gibbs-Duhem relation as M/2 = +T S + (Φq/2) + Aα′. The entropy and temperature are depend on M, q and α′, hence +dS = ∂S +∂M dM + ∂S +∂q dq + ∂S +∂α′ dα′, +(2.12) +dT = ∂T +∂M dM + ∂T +∂q dq + ∂T +∂α′ dα′. +(2.13) +We observe that dT/dα′ ≪ (dT/dM) + (dT /dq). So, for dT/dM ≫ dT/dq, we can write M ≫ q +and for dT /dM ≪ dT/dq, we can write M = q + ǫ2 where 0 < ǫ2 < 1 (near extremal case). +Therefore, the specific heat C = T (dS/dT), can be approximated as, +C = +T dS +dM +� dT +dM +�−1 , +� +dT +dM ≫ dT +dq +� +T dS +dq +� +dT +dq +�−1 +, +� +dT +dM ≪ dT +dq +� +. +(2.14) +Thus, we can express the specific heat as +C = −8πM 2 − 5 +2π +� q +M +�4 � +M 2 + 7 +5q2 +� ++ 7 +80π +� q +M +�4 +α′, +M ≫ q. +(2.15) +By considering zero limit of α′ and q, we obtain the specific heat for Schwarzschild black hole, +C = −8πM 2. However, near extremal case, we obtain +C = 4π2M 3 − 84 +√ +2π2M +5 +2 ǫ + 504π2M 2ǫ2 +− 1 +α′ +√ +2π2M +3 +2 (14720M 2 + 1827α′)ǫ3 + O(ǫ4), +M = q + ǫ2. +(2.16) +The behavior of the specific heat is plotted in Fig. 3 and Fig. 4. In the left panel of Fig. 3, we have +plotted the specific heat of this black hole in terms of the black hole mass for various values of the +correction parameter, with first condition (2.14). The dotted blue line represents the uncorrected +case. We observe that there is a phase transition point. It implies that the black hole is in the +unstable phase and its mass reduces as it emits radiation. The dashed green and the solid red lines +represent the effect of α′ correction. Here, we can observe that a second phase transition point occur +for a small mass. This is expected as α′ correction are a quantum gravitational effect. Therefore, +the final stage of the black hole is in an unstable phase. On the other hand from the right panel +of Fig. 3, we can observe that the uncharged black hole is completely in an unstable phase. The +specific heat produced by second condition of (2.14) is plotted in Fig. 4. We can observe that the +uncorrected black hole is completely stable, while the corrected black hole has a second order phase +transition. This occurs due to the divergence of the specific heat. +3 +Evaporation +As the black hole evaporates, its entropy will change. We can calculate the change in its entropy, and +use it for investigating the effect of the α′ corrections on the evaporation of such a black hole. So, +one can express the change in the quantum corrected entropy of the Reissner-Nordstr¨om black hole +as ∆S = S(rh2) − S(rh1), where S(rh1) is the initial entropy of a α′-corrected Reissner-Nordstr¨om +black hole and S(rh2) is its final entropy. Using the relation expression for the α′-corrected entropy, +we can express this change in the entropy during the evaporation process as +∆S +π += M 2 +2 +� +(1 + γ2)2 + α′ +M 2 +2 +21γ2 +2 + 18γ2 + 1 +40γ2 (1 + γ2) +� +− M 2 +1 +� +(1 + γ1)2 + α′ +M 2 +1 +21γ2 +1 + 18γ1 + 1 +40γ1 (1 + γ1) +� +, +(3.1) +– 6 – + +Figure 3. Specific heat in terms of M for +dT +dM ≫ dT +dq . +Figure 4. Specific heat in terms of M for +dT +dM ≪ dT +dq . +Here γ1 is expressed in terms of initial mass and charge (M1, q1). Similarly, γ2 is expressed in terms +of final mass and charge (M2, q2). Thus, we can write +γ1 = +� +1 − +� q1 +M1 +�2 +, γ2 = +� +1 − +� q2 +M2 +�2 +(3.2) +– 7 – + +It has been argued that the Hawking radiation might be a unitary non-thermal process, when +the self-gravitational effects of the emitted particles on the black hole geometry are not neglected +[54]. In fact, using this Parikh-Wilczek formalism [54] the black hole entropy has been can be related +to the entropy of such particles emitted [55]. Thus, we can relate the change in the entropy of the +α′-corrected black hole to the entropy of such particles, using the α′-corrected version of the Parikh- +Wilczek formalism. Even though certain problems remain in this approach [89, 90], this formalism +can be used to analyze the effect of the α′ correction on information in a black hole. Using Parikh- +Wilczek formalism [54], we first denote the number of ways in which the original classical black hole +can evaporated by emission of i particles as Ωi(M, q). Now the total number of ways in which such +a classical black hole will evaporate will be Ωt(M, q) = �∞ +i=1 Ωi(M, q). So, the probability that +the black hole will evaporated by the emission of i particles is given by Qi = Ωi(M, q)/Ωt(M, q). +Similarly, we can write the α′-corrected probability as Pi(α′) = Ωi(M, q; α′)/Ωt(M, q; α′), where +Ωi(M, q) and Ωt(M, q) are quantum gravitationally corrected Ωi(M, q) and Ωt(M, q). Here, the +quantum gravitational corrections are taken up to the order α′. +Thus, we observe that Qi = +Pi(α′)|α′=0. Using Parikh-Wilczek formalism [54, 55], entropy corresponding to the original and +α′-corrected distributions can be represented by S[Q] and S[P], such that [54, 55] +S[Q] = − +∞ +� +i=1 +Qi log Qi, +S[P] = − +∞ +� +i=1 +Pi(α′) log Pi(α′) +(3.3) +We can also write the cross entropy between the original and α′-corrected probability distributions +as [58, 59] +H[P, Q] = − +∞ +� +i=1 +Pi(α′) log Qi +(3.4) +As we are using the α′ as a state variable, we can analyze how the probability distribution changes +due to its variation. Since α′ is small, we can use the Taylor expansion of Pi(α′) near α′ = 0 and +Qi = Pi(α′)|α′=0 to obtain +H[P, Q] = − +∞ +� +i=1 +� +Pi(α′) + α′ δPi(α′) +δα′ +� +|α′=0 log Qi += S[Q] − α′ +∞ +� +i=1 +�δPi(α′) +δα′ +� +|α′=0 log Qi +(3.5) +Similarly, we can also use the Taylor expansion of Pi(α′) near α′ = 0 and write S[P] as +S[P] = S[Q] − α′ +∞ +� +i=1 +�δPi(α′) +δα′ +� +|α′=0 log Qi − α′ +∞ +� +i=1 +�δPi(α′) +δα′ +� +|α′=0 +(3.6) +The entropy and cross entropy can be used to obtain Kullback-Leibler divergence as DKL(P||Q) = +H[P, Q] − S[P], and so we can write the Kullback-Leibler divergence for these probability distribu- +tions [56, 57] +DKL(P||Q) = +∞ +� +i=1 +Pi(α′) log Pi(α′) +Qi += α′ +∞ +� +i=1 +�δPi(α′) +δα′ +� +|α′=0 +(3.7) +It measure how different the original probability distribution Qi is from the α′-corrected probability +distribution Pi(α′). So, using the expansion for entropy and cross entropy for the small α′, we +observe that the Kullback-Leibler divergence scales with α′. Thus, we observe that larger values +α′ would produce larger difference between the probability distribution of particles emitted in +Parikh-Wilczek formalism [54, 55] and the α′-corrected probability distribution. Even though the +Kullback-Leibler divergence is not a statistical distance, as it is not symmetric, it does measure +how different two distributions are from each other. Hence, it was was used to analyze the effect of +the α′ corrections on the probability distributions of emitted particles. +– 8 – + +4 +Quantum Work Distribution +We will analyze the quantum work distribution for an evaporating black hole. The quantum work +distribution has been analyzed for various different black hole solutions [40–42]. However, as quan- +tum work becomes important at short distance scales, it is important to explicitly analyze the effects +of quantum gravitational corrections on quantum work. Thus, we will analyze the effects of the α′ +corrections on quantum work distribution. We first observe that the internal energy of a Reissner- +Nordstr¨om black hole can be calculated from its entropy [91]. Thus, a change in the entropy of +a black hole will also change its internal energy. We can use the α′-corrected Reissner-Nordstr¨om +entropy to calculate the change in its internal energy as it evaporates. The internal energy of the +α′-corrected Reissner-Nordstr¨om can be written as +E = 40r14 +h (19X1 − 8X2) + 3168q8α′3 (8192X1 − X2) +15r5 +h (r4 +h − 224α′q2) (16r4 +h − 6α′q2) X1 +(4.1) +where we defined, X1 = +� +2α′q2r−4 +h ++ 8, and X2 = +� +128α′q2r−4 +h ++ 2. Now we can obtain the +change in the internal energy of this black hole as it evaporates from an initial radius rh1 to the +the final radius is rh2 as ∆E = E(rh2) − E(rh1). +At large scales, the change in the internal +energy occurs mostly because of radiation. However, as the black hole becomes sufficient small, the +average quantum work becomes important. This average quantum work is also an unitary process in +quantum thermodynamics [49, 50]. So, apart from the Parikh-Wilczek formalism [54, 55], a part of +the energy will still be lost through another unitary quantum process represented by quantum work +distribution [40–42]. Now we denote this average quantum work done by ⟨W⟩, and the energy lost +through non-thermal radiation in the Parikh-Wilczek formalism as Q as this black hole evaporates +from rh1 to rh2,[54, 55]. The change in the internal energy can only occur due to these two processes, +and so we can write ∆E = Q − ⟨W⟩ [49, 50]. +We can explicitly analyze the average quantum work for the emitted particles. As the black hole +evaporates, the energy of the particles emitted will also change. If the initial energy eigenvectors are +|Ei⟩ and the final energy eigenvectors are | ˜Ej⟩, then we can calculate quantum work distribution +by measuring the change in the Hamiltonian [45, 46]. +However, as the particles are described +by relativistic field theories, we have to use Ramsey scheme to obtain quantum work distribution +[49, 50]. Now if ρ is the density matrix for the emitted particles, then the probability for getting +energy Ei at time τ = 0 is p0 +i = |⟨Ei|ρ|Ei⟩|2. The eigenvalues of Hamiltonian evolves from |Ei⟩ to +| ˜Ej⟩ as the black hole evaporates. In Parikh-Wilczek formalism [54, 55] this evolution is represented +by a unitary operator ˆU. The conditional probability to obtain energy ˜El at τ = t, if the initial +energy was Ei at τ = 0 can be represented by pτ +l|i = ⟨ ˜Ej|U|Ei⟩|2. The difference between the initial +and the final energies is given by ¯Wi,j = ˜Ej − Ei. The probability associated with the occurrence +of this difference ¯ +W i,l can be obtained from p0 +i and pτ +l|i as +pi,j = p0 +i pτ +j|i = |⟨Ei|ρ|Ei⟩|2|⟨ ˜Ej|U|Ei⟩|2 +(4.2) +Now we represent quantum work distribution by the variable W. In absence of degeneracies, this +would coincide with ¯Wi,l. +The probability distribution associated with W can be expressed as +P(W) = � +ij pi,jδ(W − ¯ +Wi,j). Using this probability distribution, and the expression for pi,j, the +average work done can be written as +⟨W⟩ = +� � +ij +pi,jδ(W − ¯Wi,j)WdW⟨W⟩ = +� +ij +|⟨Wi|ρ|Wi⟩|⟨ ˜ +Wj|U|Wi⟩|2 � +˜Wj(τ) − Wi(0) +� +(4.3) +– 9 – + +It is possible to express this average quantum work as ⟨W⟩ = tr[ ˆH(τ)ρ(τ)] − tr[ ˆH(0)ρ(0)] [49, 50]. +The characteristic function for quantum work distribution W can be expressed as +˜P(µ) = +� +P(W)eiµW dW = ⟨eiµW ⟩ +(4.4) +where µ is a parameter which is used in the Ramsey scheme. Here, we start from an auxiliary qubit, +with |0⟩ as its ground state, and |1⟩ as its excited state. We first couple this auxiliary qubit in the +ground state to the density state ρ(0) of emitted particles at time t = 0, and write ˆρtot = ˆρ ⊗ ˆρaux. +Next we apply a Hadamard operator to the qubit. After the system has evolved from ˆH(0) to ˆH(t) +due to the evaporation for t, we can write ˆC(µ) = ˆUe−iµ ˆ +H(0) ⊗ |0⟩⟨0| + e−iµ ˆ +H(t) ˆU ⊗ |1⟩⟨1|. The +qubit state can be written as ˆρaux = TrX[ ˆC(µ)ˆρtot ˆC†(µ)], where TrX, represents a trace over all +states of emitted particles. Now we apply a second Hadamard operation to the qubit state. Using +this Hadamard operation we have extracted the information about the evolution of the system due +to the evaporation of the black hole. As we have an expression for ˆρaux, it can be compared to the +general expression for ˆρaux, which is given by 2ˆρaux = 1 + Re[ ˜P(µ)]ˆσz + Im[ ˜P(µ)]ˆσy [92, 93]. Using +this expression we obtain an expression for the characteristic function ˜P(µ). This characteristic +function can then be used to obtain the average quantum work as +⟨W⟩ = i d +dν +˜P(ν). +(4.5) +The Jarzynski equality can be used to calculate this average quantum work done as this black hole +evaporates between two states [45, 46]. This is done by relating this average quantum work to the +difference of the equilibrium free energies for black hole [43, 44] +⟨e−βW ⟩ = eβ∆F +(4.6) +Here, we have used the Jensen equality to relate the average of the exponential to the exponential +of the average as exp ⟨−βW⟩ ≤ ⟨exp (−βW)⟩. The equilibrium free energy of the α′-corrected black +hole can be written as +F = +� +960r5 +h +� +r4 +h − 224α′q2� � +16r4 +h − 6α′q2� +X1 +�−1 +× +� +− 2560r14 +h +� +3 +√ +2 − 19X1 + 8X2 +� ++ 2560q2r12 +h +� +3 +√ +2 − 253X1 + 8X2 +� ++ 960α′q2r10 +h +� +1795 +√ +2 − 5908X1 + 2056X2 +� +− 64α′q4r8 +h +� +26910 +√ +2 − 632287X1 + 30800X2 +� +− 160α′2q4r6 +h +� +4041 +√ +2 − 3160320X1 + 4632X2 +� ++ 24α′2q6r4 +h +� +17953 +√ +2 − 185185024X1 + 20568X2 +� ++ 46080α′3q6r2 +h +� +7 +√ +2 − 4096X1 + 8X2 +� +− 50688α′3q8 � +7 +√ +2 − 65536X1 + 8X2 +� � +(4.7) +Using this expression for the equilibrium free energy, we can calculate the difference between free +energies as ∆F = F(rh2)−F(rh1). In Fig. 5, we have plotted the effect of the α′ on ∆F. We observe +that α′ modify free energies and thus average quantum work for a Reissner-Nordstr¨om black hole. It +is possible to construct a microscopic model for a stretched horizon of the Reissner-Nordstr¨om black +hole, where the horizon consists of discrete constituents [94]. Using this model it is possible to write +an expression for the partition function for this black hole. The black hole partition function can +also been constructed using string theory [95]. As the black hole evaporates, the partition function +associated with the black hole will change. As the black hole evaporates from an initial partition +function Z1 to a final partition function Z2, the relative weight of these partition functions Z2/Z1 +can be related to the average quantum work as ⟨exp (−βW)⟩ = Z2/Z1 [43, 44]. Furthermore, using +– 10 – + +Figure 5. Left: ∆F in terms of M for q1 = 1 and M1 = 2. Right: ∆F in terms of the α′ for q1 = 1 and +M1 = 2. +the relation between average quantum work and equilibrium free energies, this relative weights of +the partition function can also be related to the difference between the equilibrium free energies +of the black hole as exp β∆F = Z2/Z1. Thus, we can obtain information information about the +behavior of this black hole, and its α′ correction using average quantum work. +5 +Conclusion +We have studied the effects of α′ corrections on the behavior of Reissner-Nordstr¨om black hole. The +metric for such black holes had already been constructed using heterotic superstring effective field +theory compactified on T 6 [51]. We have observed that the α′ corrections change thermodynamic +stability of a Reissner-Nordstr¨om black hole. As the thermodynamical state will depend on the +value of α′, we have used it as a thermodynamic state variable and defined a conjugate variable +corresponding to it. We have obtained a novel form of the first law using this α′ correction. We +have used Parikh-Wilczek formalism to investigated the effect of α′ corrections on non-thermal +radiation. This was done using the Kullback-Leibler divergence and cross entropy for the original +probability distribution and the α′ corrected probability distribution. The non-equilibrium quantum +thermodynamics for this α′ corrected black hole was also investigated. We have used Ramsey scheme +for emitted particles to calculate the quantum work distribution for this system. The Jarzynski +equality was used to relate average α′ corrected average quantum work to the difference of α′ +corrected free energies. This was done by using non-equilibrium quantum thermodynamics for this +black hole. As we have used α′ corrected solutions, we were able to study the effect of α′ correction +on this quantum thermodynamics of this black hole. +It would be interesting to analyze higher order α′ corrections for such black holes. We can +investigate the effects of higher order α′ corrections on the Parikh-Wilczek formalism. This can +be used to analyze how higher order quantum gravitational corrections change the non-thermal +radiation. We can also relate these quantum gravitationally corrected probability distributions, +for each other of α′ to each other. This can be done using the Kullback-Leibler divergence and +cross entropy for the each of those probability distributions. We can also investigate the effect of +such higher order α′ corrections on non-equilibrium quantum thermodynamics for this α′. We can +again use Jarzynski equality to investigate the effects of α′ corrections on the average quantum +– 11 – + +work. Furthermore, α′ corrections can be obtained for various different black hole solutions, and +these solutions can then be used to investigate the effects of such corrections on the quantum +thermodynamics of very small black holes. As the quantum work is a unitary process it would be +interesting to investigate the effect of such corrections on quantum work distribution of different +black hole solutions. +This can then be related to information paradox. +It would be possible +to study such α′ corrections for AdS black holes, and then use the Jarzynski equality to obtain +average quantum work for such corrected AdS solutions. Furthermore, as we have constructed a +novel modification of the first law, it would be interesting to combine the modification proposed in +this paper, with extended phase space thermodynamics of an AdS black hole. We can also relate +the behavior of an AdS space to its field theoretical dual. 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D 84, 026005 (2011) +– 14 – + diff --git a/l9AyT4oBgHgl3EQfyfn1/content/tmp_files/load_file.txt b/l9AyT4oBgHgl3EQfyfn1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efb7acba2e4f5dabeded3a60f1dbfaf7a3792ec7 --- /dev/null +++ b/l9AyT4oBgHgl3EQfyfn1/content/tmp_files/load_file.txt @@ -0,0 +1,850 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf,len=849 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='00687v1 [hep-th] 30 Dec 2022 Quantum Thermodynamics of an α′-Corrected Reissner-Nordstr¨om Black Hole Behnam Pourhassan,a ˙Izzet Sakallı,b Xiaoping Shi,c Mir Faizal,d Salman Sajad Wanie aSchool of Physics, Damghan University, Damghan, 3671641167, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' bPhysics Department, Eastern Mediterranean University, Famagusta 99628, North Cyprus via Mersin 10, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' c,dIrving K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Barber School of Arts and Sciences, University of British Columbia Okanagan, Kelowna, BC V1V 1V7, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' dCanadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' eDepartment of Physics Engineering, Istanbul Technical University, Istanbul, 34469 Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' E-mail: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='pourhassan@du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='ir, izzet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='sakalli@emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='tr, mirfaizalmir@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='com Abstract: In this paper, we will analyze the effects of α′ corrections on the behavior of a Reissner- Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will calculate the effects of such corrections on the thermodynamics and thermodynamic stability of such a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will also derived a novel α′-corrected first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will investigate the effect of such corrections on the Parikh-Wilczek formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This will be done using cross entropy and Kullback-Leibler divergence between the original probability distribution and the α′-corrected probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will then analyze the non-equilibrium quantum thermodynamics of this black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It will be observed that its quantum thermodynamics is corrected due to quantum gravitational corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will use Ramsey scheme for emitted particles to calculate the quantum work distribution for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The average quantum work will be related to the difference of α′-corrected free energies using the Jarzynski equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Quantum Gravitational Corrections 3 3 Evaporation 6 4 Quantum Work Distribution 9 5 Conclusion 11 1 Introduction The equilibrium thermodynamics of black holes is well suited to describe the behavior of black holes at large distances [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In this equilibrium description, temperature of the Hawking radiation is inversely proportional to its surface gravity, and the entropy of the black hole scales with its area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, as the size of the black hole reduces, its temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' At such short distances, due to high temperature, the effect of thermal fluctuations cannot be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Due to these thermal fluctuations the equilibrium description breaks down [5–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It is known that non-equilibrium quan- tum thermodynamics can be used to describe a systems where the equilibrium description breaks down [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This has motivated the use of non-equilibrium quantum thermodynamics to describe the behavior of small black holes [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In the original Bekenstein-Hawking entropy, a semi- classical approximation is used and the quantum gravitational corrections are neglected [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Even though this is well suited for large distance physics of black holes, it is important to consider quantum gravitationally corrected entropy at short distances [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the entropy of black holes is related to the holographic principle [22, 23], these quantum gravitational corrections can also be holographically obtained using the AdS/CFT correspondence [24–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The quantum gravitation- ally corrected thermodynamics for small scales black holes has been studied using string theoretical effects [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It is also possible to use properties of conformal field theory to obtain quantum gravitational corrections to black hole entropy at short distances [34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The consequences of such quantum gravitational corrections on the behavior of black branes has also been studied [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It was observed that at large distances, these quantum gravitational corrections for black branes do not change their thermodynamics, and their effects can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, their effects at short distances cannot be neglected, and they produce non-trivial modifications to the thermodynamics of black branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This is similar to the behavior of black holes [24–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The non-equilibrium quantum thermodynamics of black solutions modified by quantum gravi- tational corrections has been studied for small AdS black holes [40], Myers-Perry black hole [41], and a system of M2-M5 branes [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This quantum work distribution was expressed using the difference of quantum gravitationally corrected equilibrium free energies using the Jarzynski equality [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The Jarzynski equality is related to the quantum Crooks fluctuation theorem in non-equilibrium quantum thermodynamics [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The loss of unitarity in equilibrium black hole thermodynamics has lead to the information paradox [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, equilibrium description breaks down at short distances, and we have to use non-equilibrium quantum thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We observed that the Hawking radiation in black holes breaks unitarity as it represents heat, and heat in quantum ther- modynamics is in turn represented by a non-unitary quantum process [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, quantum – 1 – work distribution is represented by a unitary quantum process [49, 50], so at least there is a uni- tary quantum process associated with black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Furthermore, this unitary quantum process becomes important at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, it is possible that quantum thermodynamics could potentially resolve the black hole information paradox [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' At short distance it is important to consider quantum gravitational corrections, and α′ correc- tions represent such corrections in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The α′ corrections to a black hole solution have been obtained from the heterotic superstring effective action [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' These α′ corrected solutions were used to investigate the effect of these corrections on their equilibrium entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The effect of such corrections on the entropy of both BPS and non-BPS extremal dyonic black holes in heterotic string theory has also been discussed [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The near-horizon geometry of such solutions was used to calculate the entropy of these corrected black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The α′ corrected T-duality trans- formations have been used to analyze the invariance of entropy for corrected solutions [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In this paper, we will analyze the effects of α′ corrections on non-equilibrium quantum thermodynamics of a Reissner-Nordstr¨om black hole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The α′ corrections to a four dimensional Reissner- Nordstr¨om have been explicitly calculated [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This was done using heterotic superstring effective field theory compactified on T 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In fact, using this α′ corrected metric, the effect of α′ corrections on the equilibrium entropy and temperature were also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, here we will generalize this result by analyzing the effect of α′ corrections on the evaporation of this Reissner-Nordstr¨om black hole, and its quantum thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The black hole information paradox has also been addressed by considering the self-gravitation effects of the emitted particles on the black hole geometry [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using this Parikh-Wilczek formalism, it has been argued that the Hawking radiation could be a unitary non-thermal process [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In fact, using the Parikh-Wilczek formalism the black hole entropy has been related to the entropy of such emitted particles [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It is important to analyze the effects of α′ on Parikh-Wilczek formalism, as the α′ can change the behavior of black hole during its last stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, in this paper, we will investigate the effect of such quantum gravitational corrections on the Parikh-Wilczek formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will use Kullback-Leibler divergence [56, 57] and cross entropy [58, 59] to to analyze how the probability density of particles emitted during evaporation is effected by α′ corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The Kullback-Leibler divergence measures how different two probability distributions are from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Furthermore, the Kullback-Leibler divergence can be related to cross entropy and entropy of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will use these relations to obtain an expression for the effect of α′ on such probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will also use the Ramsey scheme to calculate the quantum work distribution of these emitted particles [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we will related average quantum work in quantum thermodynamics of black hole [40–42] to the quantum work distribution of emitted particles in the Parikh-Wilczek formalism [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It may be noted that the Parikh-Wilczek formalism has been used to investigate the non- thermal unitary radiation from a Reissner-Nordstr¨om black hole [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The thermodynamics of Reissner-Nordstr¨om black hole is interesting [61], as the quantum corrections to its equilibrium thermodynamics can be studied using relativistic quantum geometry [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It has been demonstrated that small thermal fluctuations to the equilibrium thermodynamics of a Reissner-Nordstr¨om black hole can modify its equilibrium entropy [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It is possible to relate these thermal fluctuations to quantum fluctuations [5–9] using the Jacobson formalism formalism [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the geometry an emergent structure in Jacobson formalism, it is possible to use it to relate the quantum fluctuation in the geometry to the thermal fluctuations in the equilibrium thermodynamics [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This formalism has been used to obtain quantum fluctuations of different black hole solutions [66–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, the Jacobson formalism [64, 65] can also be used to obtain the quantum corrections to the geometry of a Reissner-Nordstr¨om black hole from thermal fluctuations to its equilibrium thermodynamics [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The effect of a zero point length in geometry of spacetime on the thermodynamics of a Reissner-Nordstr¨om black hole have been studied using the generalized uncertainty principle [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' – 2 – The generalized uncertainty principle is a modification to the usual quantum mechanical uncertainty principle [71], and can be used to study quantum gravitational corrections to the semi-classical black hole entropy [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The generalized uncertainty principle has been motivated from string theoretical corrections [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, it is important to analyze the effects of quantum gravitational corrections on the thermodynamics of a Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, in this paper, we will explicitly analyze such effects produced by the α′ corrections on such a Reissner-Nordstr¨om black hole [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 2 Quantum Gravitational Corrections It is possible to obtain the α′ corrections to a four-dimensional Reissner-Nordstr¨om black hole using heterotic superstring effective field theory compactified on T 6 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We will analyze the quantum thermodynamics of such a four dimensional Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now this solution will be characterized by its mass M and its electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Furthermore, it will also contain an explicit α′-corrected contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The metric for such a solution can be expressed as [51], ds2 = N 2f(r)dt2 − dr2 f(r) − r2(dθ2 + sin2 θdϕ2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='1) where N 2 = 1 + α′ p2 8r4 f(r) = 1 − 2M r + p2 2r2 − α′ p2 4r4 � 1 − 3M 2r + 11p2 40r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2) and p is a physical constant related to the black hole charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' By neglecting α′ corrections, we obtain the usual uncorrected radius of the event horizon radius [75], r± = M � 1 ± � 1 − 1 2 � p M �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='3) Now, the α′-corrected radius of the event horizon is rh = r+ + α′M −2r′ +, where r′ + is defined as [51] r′ + ≡ M \uf8eb \uf8ec \uf8ed−13 20 �M p �2 + 8 5 �M p �4 − 9 80 − 21 20 � M p �2 + 8 5 � M p �4 � 1 − 1 2 � p M �2 \uf8f6 \uf8f7 \uf8f8 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='4) In the case of single electric charge q it is given by p = √ 2q [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, we will consider this case of a single charge and assume p = √ 2q for rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Here, assuming q/M ≪ 1 and α′ = aM 2, and so we can write rh = 2M +aM((M 2/5q2)−(9/80)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This represents the quantum gravitationally corrected radius of the horizon for this black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can write the corrected temperature of four dimensional Reissner-Nordstr¨om black hole as following [51], T = γ 2πM (1 + γ)2 + α′ M 2 (1 + 3γ) (1 − γ)2 160πMγ (1 + γ)5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='5) Here, for the simplicity we defined γ ≡ � 1 − (q/M)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 1 we plot the corrected temperature of a four dimensional Reissner-Nordstr¨om black hole as a function of M to analyze its behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We observe that for a large black hole, the α′ corrections do not change the temperature of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This is expected as these are quantum gravitational corrections, and should only become important at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We also observe that at intermediate length scales the system becomes – 3 – Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Temperature in terms of M with α′ = aM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' very sensitive to value of α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, at very short distances, this sensitive seems on α′ tends to reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The area of the event is Ah = 4πr2 h, while the volume of is V = 4πr3 h/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='. As the entropy of the black hole is related to its area, we can express the entropy of this black hole as [51] S = πM 2 � (1 + γ)2 + α′ M 2 21γ2 + 18γ + 1 40γ (1 + γ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='6) To analyze the effects of quantum gravitational α′ corrections on the entropy of this system, we define a µ-coefficient as the ratio of the corrected and original entropies, µS = S(α′)/S(α′ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This µ-coefficient measure the change in entropy due to α′ corrections µS = 1 + α′ M 2 21γ2 + 18γ + 1 40γ (1 + γ)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='7) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 2, we have plotted µS for various values of the correction parameter (α′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We observe that the effect of α′ corrections reduces as the size of the black hole increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This is expected as these corrections are quantum gravitational corrections, there effects can be neglected for large black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We also see that the effect produced by α′ corrections seems to converge for small values of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This seems to indicate that the effects gravitational corrections will become important at small scales, and different values of α′ could lead to similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We now observe that string tension can be taken as a variable string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In fact, the string tension has been studied in various limits [77–80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It has also been suggested that by a dynamical mechanism can be used to obtain the value of the α′ [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It has also been proposed that a bound on the α′ can be obtained from high precision experiments involving low energy quantum mechanics systems [83– 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, it is possible to view α′ as a variable and analyze its effects on thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In usual thermodynamics, different state variable their conjugate are used to analyze a thermodynamic – 4 – Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' µS in terms of M with q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This has been done for S, T and Φ, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In fact, the negative cosmological constant along with its thermodynamic conjugate has also been used as a state variable [87, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now if we take the α′ as a variable, then to specify a state in such a system, we have to specify the value of the α′ along with other thermodynamic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we can view the α′ as a thermodynamic state variable, and define A as the conjugate of the α′, such that � ∂S ∂M � q,α′ = 1 T , �∂S ∂q � M,α′ = −Φ T , � ∂S ∂α′ � M,q = −A T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='8) As the entropy is depend on the parameters M, q and α′, we can write a novel α′-corrected first law of thermodynamics as dM = T dS + Φdq + Adα′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='9) This is the modification to the first law from quantum gravitational corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We obtain the value of the corrected electrostatic potential conjugate to the black hole charge Φ as Φ = q M (1 + γ) − α′q3 80 8M 4 (1 + γ) − M 2q2 (11 + 7γ) + 3q4 M 9γ2 (1 + γ)7 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='10) Similarly, by using the α′ as a state variable, we can obtain an expression for its conjugate A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we can express A as A = −160M 6 (1 + γ) − 80q2M 4 (5 + γ) + q4M 2 (321 + 181γ) − 3q6 (27 + 7γ) 80M 7γ2 (1 + γ)6 − α′ � 16M 4 (γ − 1) + 4q2M 2 (17 − γ) + q4 (63γ − 51) � 6400M 7γ2 (1 + γ)6 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='11) – 5 – Now we can write a quantum gravitationally corrected Smarr-Gibbs-Duhem relation as M/2 = T S + (Φq/2) + Aα′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The entropy and temperature are depend on M, q and α′, hence dS = ∂S ∂M dM + ∂S ∂q dq + ∂S ∂α′ dα′, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='12) dT = ∂T ∂M dM + ∂T ∂q dq + ∂T ∂α′ dα′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='13) We observe that dT/dα′ ≪ (dT/dM) + (dT /dq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, for dT/dM ≫ dT/dq, we can write M ≫ q and for dT /dM ≪ dT/dq, we can write M = q + ǫ2 where 0 < ǫ2 < 1 (near extremal case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Therefore, the specific heat C = T (dS/dT), can be approximated as, C = T dS dM � dT dM �−1 , � dT dM ≫ dT dq � T dS dq � dT dq �−1 , � dT dM ≪ dT dq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='14) Thus, we can express the specific heat as C = −8πM 2 − 5 2π � q M �4 � M 2 + 7 5q2 � + 7 80π � q M �4 α′, M ≫ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='15) By considering zero limit of α′ and q, we obtain the specific heat for Schwarzschild black hole, C = −8πM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, near extremal case, we obtain C = 4π2M 3 − 84 √ 2π2M 5 2 ǫ + 504π2M 2ǫ2 − 1 α′ √ 2π2M 3 2 (14720M 2 + 1827α′)ǫ3 + O(ǫ4), M = q + ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='16) The behavior of the specific heat is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 3, we have plotted the specific heat of this black hole in terms of the black hole mass for various values of the correction parameter, with first condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The dotted blue line represents the uncorrected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We observe that there is a phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It implies that the black hole is in the unstable phase and its mass reduces as it emits radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The dashed green and the solid red lines represent the effect of α′ correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Here, we can observe that a second phase transition point occur for a small mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This is expected as α′ correction are a quantum gravitational effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Therefore, the final stage of the black hole is in an unstable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' On the other hand from the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 3, we can observe that the uncharged black hole is completely in an unstable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The specific heat produced by second condition of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='14) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can observe that the uncorrected black hole is completely stable, while the corrected black hole has a second order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This occurs due to the divergence of the specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 3 Evaporation As the black hole evaporates, its entropy will change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can calculate the change in its entropy, and use it for investigating the effect of the α′ corrections on the evaporation of such a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, one can express the change in the quantum corrected entropy of the Reissner-Nordstr¨om black hole as ∆S = S(rh2) − S(rh1), where S(rh1) is the initial entropy of a α′-corrected Reissner-Nordstr¨om black hole and S(rh2) is its final entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using the relation expression for the α′-corrected entropy, we can express this change in the entropy during the evaporation process as ∆S π = M 2 2 � (1 + γ2)2 + α′ M 2 2 21γ2 2 + 18γ2 + 1 40γ2 (1 + γ2) � − M 2 1 � (1 + γ1)2 + α′ M 2 1 21γ2 1 + 18γ1 + 1 40γ1 (1 + γ1) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='1) – 6 – Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Specific heat in terms of M for dT dM ≫ dT dq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Specific heat in terms of M for dT dM ≪ dT dq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Here γ1 is expressed in terms of initial mass and charge (M1, q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Similarly, γ2 is expressed in terms of final mass and charge (M2, q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we can write γ1 = � 1 − � q1 M1 �2 , γ2 = � 1 − � q2 M2 �2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2) – 7 – It has been argued that the Hawking radiation might be a unitary non-thermal process, when the self-gravitational effects of the emitted particles on the black hole geometry are not neglected [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In fact, using this Parikh-Wilczek formalism [54] the black hole entropy has been can be related to the entropy of such particles emitted [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we can relate the change in the entropy of the α′-corrected black hole to the entropy of such particles, using the α′-corrected version of the Parikh- Wilczek formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Even though certain problems remain in this approach [89, 90], this formalism can be used to analyze the effect of the α′ correction on information in a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using Parikh- Wilczek formalism [54], we first denote the number of ways in which the original classical black hole can evaporated by emission of i particles as Ωi(M, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now the total number of ways in which such a classical black hole will evaporate will be Ωt(M, q) = �∞ i=1 Ωi(M, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, the probability that the black hole will evaporated by the emission of i particles is given by Qi = Ωi(M, q)/Ωt(M, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Similarly, we can write the α′-corrected probability as Pi(α′) = Ωi(M, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' α′)/Ωt(M, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' α′), where Ωi(M, q) and Ωt(M, q) are quantum gravitationally corrected Ωi(M, q) and Ωt(M, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Here, the quantum gravitational corrections are taken up to the order α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we observe that Qi = Pi(α′)|α′=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using Parikh-Wilczek formalism [54, 55], entropy corresponding to the original and α′-corrected distributions can be represented by S[Q] and S[P], such that [54, 55] S[Q] = − ∞ � i=1 Qi log Qi, S[P] = − ∞ � i=1 Pi(α′) log Pi(α′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='3) We can also write the cross entropy between the original and α′-corrected probability distributions as [58, 59] H[P, Q] = − ∞ � i=1 Pi(α′) log Qi (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='4) As we are using the α′ as a state variable, we can analyze how the probability distribution changes due to its variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Since α′ is small, we can use the Taylor expansion of Pi(α′) near α′ = 0 and Qi = Pi(α′)|α′=0 to obtain H[P, Q] = − ∞ � i=1 � Pi(α′) + α′ δPi(α′) δα′ � |α′=0 log Qi = S[Q] − α′ ∞ � i=1 �δPi(α′) δα′ � |α′=0 log Qi (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='5) Similarly, we can also use the Taylor expansion of Pi(α′) near α′ = 0 and write S[P] as S[P] = S[Q] − α′ ∞ � i=1 �δPi(α′) δα′ � |α′=0 log Qi − α′ ∞ � i=1 �δPi(α′) δα′ � |α′=0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='6) The entropy and cross entropy can be used to obtain Kullback-Leibler divergence as DKL(P||Q) = H[P, Q] − S[P], and so we can write the Kullback-Leibler divergence for these probability distribu- tions [56, 57] DKL(P||Q) = ∞ � i=1 Pi(α′) log Pi(α′) Qi = α′ ∞ � i=1 �δPi(α′) δα′ � |α′=0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='7) It measure how different the original probability distribution Qi is from the α′-corrected probability distribution Pi(α′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, using the expansion for entropy and cross entropy for the small α′, we observe that the Kullback-Leibler divergence scales with α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we observe that larger values α′ would produce larger difference between the probability distribution of particles emitted in Parikh-Wilczek formalism [54, 55] and the α′-corrected probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Even though the Kullback-Leibler divergence is not a statistical distance, as it is not symmetric, it does measure how different two distributions are from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Hence, it was was used to analyze the effect of the α′ corrections on the probability distributions of emitted particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' – 8 – 4 Quantum Work Distribution We will analyze the quantum work distribution for an evaporating black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The quantum work distribution has been analyzed for various different black hole solutions [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, as quan- tum work becomes important at short distance scales, it is important to explicitly analyze the effects of quantum gravitational corrections on quantum work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we will analyze the effects of the α′ corrections on quantum work distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We first observe that the internal energy of a Reissner- Nordstr¨om black hole can be calculated from its entropy [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, a change in the entropy of a black hole will also change its internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can use the α′-corrected Reissner-Nordstr¨om entropy to calculate the change in its internal energy as it evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The internal energy of the α′-corrected Reissner-Nordstr¨om can be written as E = 40r14 h (19X1 − 8X2) + 3168q8α′3 (8192X1 − X2) 15r5 h (r4 h − 224α′q2) (16r4 h − 6α′q2) X1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='1) where we defined, X1 = � 2α′q2r−4 h + 8, and X2 = � 128α′q2r−4 h + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now we can obtain the change in the internal energy of this black hole as it evaporates from an initial radius rh1 to the the final radius is rh2 as ∆E = E(rh2) − E(rh1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' At large scales, the change in the internal energy occurs mostly because of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, as the black hole becomes sufficient small, the average quantum work becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This average quantum work is also an unitary process in quantum thermodynamics [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' So, apart from the Parikh-Wilczek formalism [54, 55], a part of the energy will still be lost through another unitary quantum process represented by quantum work distribution [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now we denote this average quantum work done by ⟨W⟩, and the energy lost through non-thermal radiation in the Parikh-Wilczek formalism as Q as this black hole evaporates from rh1 to rh2,[54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The change in the internal energy can only occur due to these two processes, and so we can write ∆E = Q − ⟨W⟩ [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can explicitly analyze the average quantum work for the emitted particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the black hole evaporates, the energy of the particles emitted will also change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' If the initial energy eigenvectors are |Ei⟩ and the final energy eigenvectors are | ˜Ej⟩, then we can calculate quantum work distribution by measuring the change in the Hamiltonian [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' However, as the particles are described by relativistic field theories, we have to use Ramsey scheme to obtain quantum work distribution [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now if ρ is the density matrix for the emitted particles, then the probability for getting energy Ei at time τ = 0 is p0 i = |⟨Ei|ρ|Ei⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The eigenvalues of Hamiltonian evolves from |Ei⟩ to | ˜Ej⟩ as the black hole evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In Parikh-Wilczek formalism [54, 55] this evolution is represented by a unitary operator ˆU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The conditional probability to obtain energy ˜El at τ = t, if the initial energy was Ei at τ = 0 can be represented by pτ l|i = ⟨ ˜Ej|U|Ei⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The difference between the initial and the final energies is given by ¯Wi,j = ˜Ej − Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The probability associated with the occurrence of this difference ¯ W i,l can be obtained from p0 i and pτ l|i as pi,j = p0 i pτ j|i = |⟨Ei|ρ|Ei⟩|2|⟨ ˜Ej|U|Ei⟩|2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2) Now we represent quantum work distribution by the variable W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In absence of degeneracies, this would coincide with ¯Wi,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The probability distribution associated with W can be expressed as P(W) = � ij pi,jδ(W − ¯ Wi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using this probability distribution, and the expression for pi,j, the average work done can be written as ⟨W⟩ = � � ij pi,jδ(W − ¯Wi,j)WdW⟨W⟩ = � ij |⟨Wi|ρ|Wi⟩|⟨ ˜ Wj|U|Wi⟩|2 � ˜Wj(τ) − Wi(0) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='3) – 9 – It is possible to express this average quantum work as ⟨W⟩ = tr[ ˆH(τ)ρ(τ)] − tr[ ˆH(0)ρ(0)] [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The characteristic function for quantum work distribution W can be expressed as ˜P(µ) = � P(W)eiµW dW = ⟨eiµW ⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='4) where µ is a parameter which is used in the Ramsey scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Here, we start from an auxiliary qubit, with |0⟩ as its ground state, and |1⟩ as its excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We first couple this auxiliary qubit in the ground state to the density state ρ(0) of emitted particles at time t = 0, and write ˆρtot = ˆρ ⊗ ˆρaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Next we apply a Hadamard operator to the qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' After the system has evolved from ˆH(0) to ˆH(t) due to the evaporation for t, we can write ˆC(µ) = ˆUe−iµ ˆ H(0) ⊗ |0⟩⟨0| + e−iµ ˆ H(t) ˆU ⊗ |1⟩⟨1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The qubit state can be written as ˆρaux = TrX[ ˆC(µ)ˆρtot ˆC†(µ)], where TrX, represents a trace over all states of emitted particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Now we apply a second Hadamard operation to the qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using this Hadamard operation we have extracted the information about the evolution of the system due to the evaporation of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As we have an expression for ˆρaux, it can be compared to the general expression for ˆρaux, which is given by 2ˆρaux = 1 + Re[ ˜P(µ)]ˆσz + Im[ ˜P(µ)]ˆσy [92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using this expression we obtain an expression for the characteristic function ˜P(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This characteristic function can then be used to obtain the average quantum work as ⟨W⟩ = i d dν ˜P(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='5) The Jarzynski equality can be used to calculate this average quantum work done as this black hole evaporates between two states [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This is done by relating this average quantum work to the difference of the equilibrium free energies for black hole [43, 44] ⟨e−βW ⟩ = eβ∆F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='6) Here, we have used the Jensen equality to relate the average of the exponential to the exponential of the average as exp ⟨−βW⟩ ≤ ⟨exp (−βW)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The equilibrium free energy of the α′-corrected black ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='hole can be written as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='F = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='960r5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='r4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h − 224α′q2� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='16r4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h − 6α′q2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='X1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='− 2560r14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 19X1 + 8X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='+ 2560q2r12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 253X1 + 8X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='+ 960α′q2r10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='1795 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 5908X1 + 2056X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='− 64α′q4r8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='26910 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 632287X1 + 30800X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='− 160α′2q4r6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='4041 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 3160320X1 + 4632X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='+ 24α′2q6r4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='17953 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 185185024X1 + 20568X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='+ 46080α′3q6r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 4096X1 + 8X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='− 50688α′3q8 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='2 − 65536X1 + 8X2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content='7) Using this expression for the equilibrium free energy, we can calculate the difference between free energies as ∆F = F(rh2)−F(rh1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 5, we have plotted the effect of the α′ on ∆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We observe that α′ modify free energies and thus average quantum work for a Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It is possible to construct a microscopic model for a stretched horizon of the Reissner-Nordstr¨om black hole, where the horizon consists of discrete constituents [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Using this model it is possible to write an expression for the partition function for this black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The black hole partition function can also been constructed using string theory [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the black hole evaporates, the partition function associated with the black hole will change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the black hole evaporates from an initial partition function Z1 to a final partition function Z2, the relative weight of these partition functions Z2/Z1 can be related to the average quantum work as ⟨exp (−βW)⟩ = Z2/Z1 [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Furthermore, using – 10 – Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Left: ∆F in terms of M for q1 = 1 and M1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Right: ∆F in terms of the α′ for q1 = 1 and M1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' the relation between average quantum work and equilibrium free energies, this relative weights of the partition function can also be related to the difference between the equilibrium free energies of the black hole as exp β∆F = Z2/Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Thus, we can obtain information information about the behavior of this black hole, and its α′ correction using average quantum work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' 5 Conclusion We have studied the effects of α′ corrections on the behavior of Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The metric for such black holes had already been constructed using heterotic superstring effective field theory compactified on T 6 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We have observed that the α′ corrections change thermodynamic stability of a Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the thermodynamical state will depend on the value of α′, we have used it as a thermodynamic state variable and defined a conjugate variable corresponding to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We have obtained a novel form of the first law using this α′ correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We have used Parikh-Wilczek formalism to investigated the effect of α′ corrections on non-thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This was done using the Kullback-Leibler divergence and cross entropy for the original probability distribution and the α′ corrected probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The non-equilibrium quantum thermodynamics for this α′ corrected black hole was also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We have used Ramsey scheme for emitted particles to calculate the quantum work distribution for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' The Jarzynski equality was used to relate average α′ corrected average quantum work to the difference of α′ corrected free energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This was done by using non-equilibrium quantum thermodynamics for this black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As we have used α′ corrected solutions, we were able to study the effect of α′ correction on this quantum thermodynamics of this black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It would be interesting to analyze higher order α′ corrections for such black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can investigate the effects of higher order α′ corrections on the Parikh-Wilczek formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This can be used to analyze how higher order quantum gravitational corrections change the non-thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can also relate these quantum gravitationally corrected probability distributions, for each other of α′ to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This can be done using the Kullback-Leibler divergence and cross entropy for the each of those probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can also investigate the effect of such higher order α′ corrections on non-equilibrium quantum thermodynamics for this α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' We can again use Jarzynski equality to investigate the effects of α′ corrections on the average quantum – 11 – work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Furthermore, α′ corrections can be obtained for various different black hole solutions, and these solutions can then be used to investigate the effects of such corrections on the quantum thermodynamics of very small black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' As the quantum work is a unitary process it would be interesting to investigate the effect of such corrections on quantum work distribution of different black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' This can then be related to information paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' It would be possible to study such α′ corrections for AdS black holes, and then use the Jarzynski equality to obtain average quantum work for such corrected AdS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfyfn1/content/2301.00687v1.pdf'} +page_content=' Furthermore, as we have constructed a novel modification of the first law, it would be interesting to combine the modification proposed in this paper, with extended phase space thermodynamics of an AdS black 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Klevs1,2 · F. Stefani1 · L. Jouve3 +© Springer •••• +Abstract We consider a conventional α − Ω-dynamo model with meridional +circulation that exhibits typical features of the solar dynamo, including a Hale +cycle period of around 20 years and a reasonable shape of the butterfly diagram. +With regard to recent ideas of a tidal synchronization of the solar cycle, we +complement this model by an additional time-periodic α-term that is localized +in the tachocline region. It is shown that amplitudes of some dm/s are sufficient +for this α-term to become capable of entraining the underlying dynamo. We +argue that such amplitudes of α may indeed be realistic, since velocities in the +range of m/s are reachable, e.g., for tidally excited magneto-Rossby waves. +Keywords: Solar cycle, Models Helicity, Theory +1. Introduction +The general idea that solar activity variations might be linked to the orbital +motion of the planets traces back to Wolf (1859), and was kept alive, through- +out one and a half centuries, by a number of authors (de la Rue et al., 1872; +Bollinger, 1952; Jose, 1965; Takahashi, 1968; Wood, 1972; ¨Opik, 1972; Condon +and Schmidt, 1975; Charvatova, 1997; Zaqarashvili, 1997; Landscheidt, 1999; +Palus et al., 2000; De Jager and Versteegh, 2005; Wolff and Patrone, 2010; +Abreu et al., 2012; Callebaut, de Jager, and Duhau, 2012). The more specific +coincidence, though, of the 11.07-year alignment cycle of the tidally dominant +planets Venus, Earth and Jupiter with the Schwabe cycle was brought to the +� F. Stefani +f.stefani@hzdr.de +1 +Helmholtz-Zentrum Dresden – Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, +Germany +2 +University of Latvia, Institute for Numerical Modelling, 3 Jelgavas street, Riga, +LV-1004, Latvia +3 +Univ. Toulouse, IRAP, CNRS, UMR 5277,CNES,UPS, F-31400 Toulouse, France +SOLA: klevs.tex; 16 January 2023; 1:27; p. 1 +arXiv:2301.05452v1 [astro-ph.SR] 13 Jan 2023 + +M. Klevs et al. +fore only recently by Hung (2007); Scafetta (2012); Wilson (2013); Okhlopkov +(2016). +Since even such a remarkable agreement between the average values of two +periods might still be a pure coincidence, the question of whether there is a +phase coherence between the two time series becomes of utmost importance. +The possible phase stability of the Schwabe cycle was first discussed in the paper +“Is there a chronometer hidden deep in the Sun?” by Dicke (1978). Analyzing +the ratio between the mean square of the residuals (i.e., the distances between +the instants of the actual cycle maxima and the hypothetical maxima according +to a linear trend) to the mean square of the differences between two consec- +utive residuals, Dicke’s conclusions favoured a clocked process over a random +walk process. However, apart from the poor statistics connected with the mere +25 maxima taken into account, one should also take seriously Hoyng’s later +warning (Hoyng, 1996) that any α-quenching mechanism could easily lead to a +sort of self-stabilization of the solar dynamo, making a genuine random walk +process “disguise” itself as a clocked process - at least for some centuries. A +complementary type of cycle stability appears as a typical feature of conven- +tional Babcock-Leighton dynamos whose period is largely determined by the +turnover time of the meridional circulation (Dikpati and Charbonneau, 1999; +Charbonneau and Dikpati, 2000; Charbonneau, 2020), which is indeed assumed +to be much less fluctuating than the α effect in the convection zone. +With those caveats in mind, we had recently re-considered (Stefani et al., +2020b) the longer time series of cycle minima/maxima as bequeathed to us by +Schove (1983), and matched them with two series of the cosmogenic isotopes +10Be and 14C. Apart from the hardly decidable existence, or not, of two “lost +cycles” (or phase jumps) around 1563 (Link, 1978) and 1795 (Usoskin et al., +2002), our analysis confirmed, by and large, Dicke’s conclusion in favour of a +clocked cycle, now throughout the last millennium. This result was then put +into the context of the most remarkable, though widely overlooked, work of Vos +et al. (2004) whose analysis of two series of algae-related data from 10000-9000 +cal. BP had evidenced a phase-stable Schwabe cycle with a period of 11.04 years. +In view of those two independent thousand-year long segments showing nearly +identical Schwabe cycles with average periods between 11.04 and 11.07 years +and the strong evidence for phase stability in either case, we consider it at least +worthwhile to quest for a possible physical mechanism that could be capable of +linking the weak tidal forces as exerted by planets with the solar dynamo. Setting +out from the numerical observation (Weber et al., 2013, 2015; Stefani et al., 2016) +that a tide-like influence (with its typical m = 2 azimuthal dependence) can +entrain the helicity oscillation of an underlying m = 1 instability (the Tayler +instability (Tayler, 1973; Seilmayer et al., 2012) for that matter) with barely +changing its energy content, we have pursued some rudimentary synchronization +studies in the framework of simple 0D and 1D α − Ω-dynamo models (Stefani +et al., 2017, 2018; Stefani, Giesecke and Weier, 2019). Within the same frame- +work, we recently tried (Stefani et al., 2020a; Stefani, Stepanov and Weier, 2021) +to explain also the longer term Suess-de Vries cycle in terms of a beat period +(Wilson, 2013; Solheim, 2013) between the fundamental 22.14-year Hale cycle +and the 19.86-yr period of the Sun’s barycentric motion (forced, in turn, by the +SOLA: klevs.tex; 16 January 2023; 1:27; p. 2 + +A synchronized two-dimensional α − Ω model of the solar dynamo +orbits of Jupiter and Saturn (Cionco and Pavlov, 2018)) . With the intervening +spin-orbit coupling remaining poorly understood, we took resort to the same +buoyancy instability mechanism as it had been been employed by Abreu et al. +(2012) to plausibilize typical modulation periods on the centennial time-scale. +Yet, this similarity between the final results notwithstanding, the fundamental +time-scales of our model (22.14 and 19.86 years) that generate the much longer +beat period of 193 years, are still close to the period of the undisturbed dynamo. +Our mechanism for explaining long-term modulations might, therefore, be less +vulnerable to stochastic noise than what was discussed by Charbonneau (2022) +in relation to the original model of Abreu et al. (2012). +Admittedly, being restricted to the latitudinal coordinate, our simple 1D +dynamo model did not have the requisite level of detail to give a quantitative +answer to Charbonneau’s recent question of “what, then, can be considered a +physically reasonable amplitude for external forcing” (Charbonneau, 2022). It +was all the more encouraging that, utilizing a 2D Babcock-Leighton model with +a periodic perturbation of the lower operating field threshold of the source term, +Charbonneau (2022) found a similarly robust synchronization mechanism as +Stefani, Giesecke and Weier (2019). Such a variation of the lower operating field +threshold would correspond to variations of the field loss term κ as employed in +(Stefani et al., 2020a; Stefani, Stepanov and Weier, 2021) to parameterize the +spin-orbit coupling with its 19.86-yr periodicity. While we do not exclude a viable +physical translation of the (11.07-yr periodic) tidal forcing into such a type of +variation of the field storage capacity, in this paper we will stick to our original +idea that it is essentially the α effect that is affected by the tides. Specifically, +we seek to know then how much of this periodic α variation would be needed to +accomplish synchronization of an otherwise conventional α−Ω-dynamo. Guided +by a rough estimation based on the virial assumption Upot ≈ Ekin, we consider +approximately 1 m/s an upper limit for the tide-induced velocity variation. Given +that the value of α, which reflects only the helical part of the turbulence, is +typically one order of magnitude lower than the underlying velocity, the focus +of our modelling will be on whether α-values of the order of dm/s are sufficient +to entrain the entire solar dynamo. +To answer this very specific question, we step back from the more sophisti- +cated double-synchronization model of Stefani et al. (2020a); Stefani, Stepanov +and Weier (2021) and restrict ourselves to the very basic tidal synchronization +of the Schwabe/Hale cycle. In the next section, we present a rather conven- +tional two-dimensional α − Ω-dynamo with meridional circulation up, utilizing +observation-constrained values for Ω and up, and employing more or less realistic +values of α and the magnetic diffusivity η. To keep the model simple, no specific +Babcock-Leighton source term is added to the α-effect “living” in the convection +zone. In the next section, we first adjust the value of η to provide a reasonable +natural period of the undisturbed dynamo. While the most simple form of the +α − Ω model leads, as usual, to a badly shaped butterfly diagram, the correct +butterfly shape is recovered by switching on the meridional circulation. Based +on the reference model thus defined, we will then assess in detail how much α +variation in the tachocline region is actually needed for synchronization. +The paper will conclude with a short discussion of the results and some +prospects for future work. +SOLA: klevs.tex; 16 January 2023; 1:27; p. 3 + +M. Klevs et al. +2. The model +In this section, we motivate and describe our mean-field solar dynamo model and +discuss its numerical implementation. Considering only axi-symmetric solutions, +we work with a system of partial differential equations whose spatial variables are +the co-latitude and the radius. Intentionally, the model has been kept similarly +simple as the benchmark model of Jouve et al. (2008). +As usual, the magnetic field is split into a poloidal component BP (r, Θ, t) = +∇ × (A(r, Θ, t)eφ) and a toroidal component BT (r, Θ, t) = B(r, Θ, t)eφ. The +main sources of dynamo action are the gradient of the angular velocity Ω and +the α-effect resulting from the helical part of the turbulence in the convection +zone. While our model is not a Babcock-Leighton model (which would require +a particular source term at the surface) it is a flux-transport model in that it +comprises a meridional circulation up, mainly to ensure a realistic shape of the +butterfly diagram. +Choosing the solar radius R⊙ = 695700 km as the length and the diffusive +time R2 +⊙/ηt as the time scale, we employ here - as in Jouve et al. (2008) - +the dimensionless form of the coupled induction equations for the azimuthal +components B := Bφ of the magnetic field and A := Aφ of the vector potential, +∂B +∂t += ˜ηD2B + 1 +s +∂(sB) +∂r +∂˜η +∂r − Rmsup · ∇ +�B +s +� ++ CΩs(∇ × (Aeφ)) · ∇Ω (1) +∂A +∂t += ˜ηD2A − Rm +s up · ∇(sA) + Cc +ααcB + Cp +ααpB, +(2) +wherein we use the notations D2 := (∇2 − s−2), s := r sin θ and ˜η = η/ηt, with +ηt being the turbulent magnetic diffusivity in the convection zone. +This system is governed by four magnetic Reynolds numbers characterizing, +respectively, the effects of shear, meridional circulation, and two different α +terms: +CΩ = ΩeqR2 +⊙/ηt +(3) +Rm = u0R⊙/ηt +(4) +Cc +α = αc +maxR⊙/ηt +(5) +Cp +α = αp +maxR⊙/ηt . +(6) +Herein, Ωeq = 2π × 456 nHz is the angular velocity at the equator, and u0 and +αc +max and αp +max are the typical intensities of the meridional circulation and the +two separate α effects in the convection zone and in the tachocline region. In con- +trast to Guerrero and de Gouveia Dal Pino (2007); Jouve et al. (2008); Sanchez +et al. (2014), we do not incorporate any specific Babcock-Leighton source term. +We suppose the turbulent magnetic diffusivity ηt in the convection zone to +be dominated by a strong β effect, whereas it is much smaller in the relatively +quiet tachocline region. Refraining from more complicated structures of η as +employed, e.g., in Guerrero and de Gouveia Dal Pino (2007) or Sanchez et al. +SOLA: klevs.tex; 16 January 2023; 1:27; p. 4 + +A synchronized two-dimensional α − Ω model of the solar dynamo +Figure 1. Spatial structures of the main ingredients of the dynamo model in the meridional +plane. (a) Isolines of Ω(r, Θ)/Ωmax. (b) Streamlines of up(r, Θ). (c) Constant part of α, taken +in the unquenched state: αc(r, Θ)/αc +max. (d) Periodic part of α, with the resonance term set +to 1: αp(r, Θ)/αp +max. +(2014), we use here the simple form of Jouve et al. (2008) +˜η(r) = ηc +ηt ++ 1 +2 +� +1 − ηc +ηt +� � +1 + erf +�r − rc +d +�� +(7) +with ηc = 0.01ηt, rc = 0.7 and d = 0.02, which shows a smoothed-out jump (by +a factor of 100) between the radiation zone and the convection zone. +For the angular velocity we apply the same spatial structure as in Jouve et al. +(2008): +Ω(r, Θ) = CΩ +� +Ωc + 1 +2 +� +1 + erf +�r − rc +d +�� +(1 − Ωc − c2 cos2 Θ) +� +(8) +with rc = 0.7, d = 0.02, Ωc = 0.92 and c2 = 0.2 (see Figure 1(a)). +For the meridional circulation we chose, again as in Jouve et al. (2008), one +single cell defined by up = ∇ × (ψ(r, Θ)eφ) with the stream function +ψ(r, Θ) = Rm +� +− 2 +π +(r − rb)2 +(1 − rb) sin +� +π r − rb +1 − rb +� +cos Θ sin Θ +� +(9) +with rb = 0.65 (see Figure 1(b)). We are well aware of the fact that the specific +structure of up is much less settled than that of Ω(r, Θ), and that more compli- +cated two-cell flows (Kosovichev et al., 2022) might also be considered in future +improvements of our model. +Finally, α = αc + αp is thought to consist of a conventional part αc in the +convection zone, whose time-dependence stems only from the quenching by the +magnetic field, +αc(r, Θ, t) = Cc +α +3 +√ +3 +4 +sin2 Θ cos Θ +� +1 + erf +�r − rc +d +�� � +1 + |B(r, Θ, t)|2 +B2 +0 +�−1 +(10) +SOLA: klevs.tex; 16 January 2023; 1:27; p. 5 + +90° +.06 +.06 +90° +45° +45° +45° +1.00 +0.75 +0.50 +0.25 +0° +0° +0° +0° +0.00 +-0.25 +-0.50 +-0.75 +0.65 +0.65 +0.65 +0.65 +-45° +45° +45° +-1.00 +1.00 +1.00 +1.00 +1.00 +-90° +-90° +-90° +.06- +(a) +(b) +(c) +(d)M. Klevs et al. +with B0 = 1, and an explicitly time-dependent (with forcing period Tf) part αp +that is concentrated in the tachocline region, +αp(r, Θ, t) = Cp +α +1 +√ +2 sin2 Θ cos Θ +� +1 + erf +�r − rc +d +�� � +1 − erf +�r − rd +d +�� +× +× +|B(r, Θ, t)|2 +1 + |B(r, Θ, t)|4 sin(2πt/Tf) , +(11) +where rd = 0.75. Note that the factor on the second line of Eq. (11) represents +a resonance term as introduced in Stefani et al. (2016) in order to account +for a field-dependent optimal reaction of the underlying instability (e.g., Tayler +instability) on the tidal forcing. A similar field dependence has been used, e.g., +in Charbonneau (2022), although with the slightly different interpretation as a +nonlinearity of the non-local source term that incorporates both a lower and +upper operating threshold on the strength of the toroidal magnetic at the base +of the convection zone. The spatial structures of these two α terms are visualized +in Figure 1(c,d), in either case disregarding any magnetic-field dependence. +For the numerical solution, an explicit finite difference scheme in two di- +mensions in spherical coordinates is used. As in R¨udiger et al. (2003), the +standard resolution was 64 × 64 grid points in both radial and latitudinal di- +rections. The equations are solved with perfect conductor boundary conditions +A = ∂(rB)/∂r = 0 at r = 0.65R⊙ and vertical field conditions Bφ = BΘ = 0 at +r = R⊙. +3. Results +In this section we present and assess the results of three dynamo models with +increasing complexity. +3.1. Non-synchronized model, without meridional circulation +First we consider the simplest case of a Parker’s migratory dynamo (Parker, +1955), without any synchronization term (αp = 0), and without meridional +circulation (up = 0). For the sake of concreteness, we set ηt = 2.13×1011 cm2/s, +and αc +max = 1.30 m/s, which both are close to the respective geometric means of +the lower and upper values as typically found in the literature (1010 −1013 cm2/s +for η and 10 − 103 cm/s for α, see Charbonneau (2020)). The resulting magnetic +Reynolds numbers according to Equations (3) and (5) are CΩ = 65100 and +Cc +α = 42.46. The radial dependencies of η(r) and αc (in its unquenched form) +are illustrated, for Θ = 45◦, in Figure 2(a). Note that at this particular angle +αc(r) does not reach the maximum value of 1.30 m/s. +Figure 3 illustrates the resulting field dependence on time and latitude, taken +at r = 0.95, showing a reasonable dynamo cycle period of Td = 14.27 years (i.e. +0.0198 diffusion times), but a badly shaped butterfly diagram. +SOLA: klevs.tex; 16 January 2023; 1:27; p. 6 + +A synchronized two-dimensional α − Ω model of the solar dynamo +Figure 2. Radial dependence of various dynamo ingredients in physical units, all taken at +Θ = 45◦. (a) Diffusivity η(r) (black), αc(r) in the unquenched form (violet), and αp(r) for +αp +max = αc +max and with the field-dependent resonance factor artificially set to 1 (red). (b) +uΘ(r) resulting from the stream function of Eq. (9) for three different Rm. +Figure 3. Contour-plots BΘ(r = 0.95, θ, t), Bφ(r = 0.7, θ, t), and Br(r = 0.95, θ, t) and of +|B(r = 0.95, θ, t)| for the non-synchronized model without meridional circulation. Note that +the ordinate axis represents not the colatitude θ but the normal solar latitude 90◦ − θ. +3.2. Non-synchronized model, with meridional circulation +In order to recover the correct shape of the butterfly diagram, we switch on a +meridional circulation, setting its value to u0 = 5.2 m/s, which corresponds to +Rm = 170. For this value, as well as for Rm = 200 and 240, the radial dependence +of uΘ is shown, again for Θ = 45◦, in Figure 2(b). While the values uΘ at r = 1 +are by factor of appr. two too low compared with observations, the typical values +of 1-2 m/s at the base of the convection zone are quite compatible with values +from helioseismology. Actually, the latter velocities are the crucial ones to set +the cycle period. +As seen in Figure 4 we obtain now a butterfly diagram of rather decent shape, +and a slightly changed cycle period of Td = 22.798 years. This will serve in the fol- +SOLA: klevs.tex; 16 January 2023; 1:27; p. 7 + +(a) +(b) +3 +2.2 +1.4 +2 +2 +1.2 +1 +1.8 +0 +1.6 +1 +-1 +1.4 +[s/w] +S +-2 +0.8 +1.2 +[m/ +-3 +Q +1 +n +0.6 +0.8 +-4 +uα for Rm=170 +0.4 +0.6 +-5 +Qb +Rm=200 +0.4 +-6 +Rm=240 +0.2 +0.2 +-7 +0 +0 +-8 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0.65 +0.7 +0.75 +0.80.85 +0.9 +0.95 +r.r=0.95 +90 +latitude, [deg] +0.25 +45 +0.00 +45 +0.25 +-90 +B, r=0.7 +latitude, [deg] +90 +10 +45 +0 +0 +45 +-10 +90 +Br, r=0.95 +90 +latitude, [deg] +0.2 +45 +0.0 +0 +45 +-0.2 +90 +90 +latitude, [deg] +45 +3 +0 +45 +90 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +t, [yr]M. Klevs et al. +Figure 4. Same as Figure 3, but including meridional circulation with Rm = 170. +lowing as the reference dynamo model whose synchronization is to be evaluated +thereupon. While further improvements of the spatio-temporal features of the +magnetic field are certainly possible (for example, when including an appropriate +Babcock-Leighton source term), we refrain from any further sophistication of the +model. +3.3. Synchronized model +Finally, we switch on the periodic α term with an assumed forcing period of +Tf = 11.00 years (we do not insist here on the precise value of 11.07 years). +The radial dependence of αp is illustrated by the red curve in Figure 2(a). Note, +however, that here αp +max has the same value of 1.30 m/s as the corresponding +αc +max, and that the field-dependent resonance term in Equation (11) is artificially +set to 1. In reality, the resonance term reduces this value by a factor of 2 for the +optimum field strength, and even more so outside the optimum. +As shown in Figure 5, for the specific value αp +max = 0.52 m/s we obtain now +the dynamo period Td = 22.00 years which corresponds to twice the period Tf +of the forcing. +In Figure 6 we plot the dependence of the dynamo period Td on αp +max. Here we +have used a couple of ratios of the “natural” period Tn (of the non-synchronized +dynamo with αp +max = 0) to the forcing periods Tf by simply changing the am- +plitude of meridional circulation which governs Tn. Very similar to Figure 10 in +Stefani, Giesecke and Weier (2019), and to Figure 10 in Charbonneau (2022), we +obtain a clear parametric resonance for some critical value of αp +max that depends +on the initial distance between twice the forcing period Tf and the natural +period Tn of the unperturbed dynamo. As we had chosen αc +max = 1.30 m/s, +synchronization occurs for an amplitude of αp +max in the range of some dm/s. The +relative smallness of this number is, of course, a consequence of the 100 times +smaller value of η in the tachocline region which amplifies correspondingly the +SOLA: klevs.tex; 16 January 2023; 1:27; p. 8 + +Be, r=0.95 +90 +latitude, [deg] +> +0.25 +0.00 +45 +-0.25 +90 +Bs, r=0.7 +90 +latitude, [deg] +10 +45 +0 +0 +45 +-10 +90 +Br, r=0.95 +latitude, [deg] +90 +45 +0 +0 +45 +90 +Bl, r=0.95 +latitude, [deg] +90 +6 +4 +45 +2 +90 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +t, [yr]A synchronized two-dimensional α − Ω model of the solar dynamo +Figure 5. Same as Figure 4, but with synchronization by a periodic α-term with amplitude +αp +max = 0.52 m/s and period Tf = 11.00 years. +induction effect of αp, even if the latter is concentrated in a significantly smaller +zone than αc. That said, we must also admit that synchronization requires a +certain proximity of 2Tf and Tn; for the Rm values indicated by the dashed lines +in Figure 6 no clear synchronization was observed even for the highest considered +value of αp +max/αc +max = 1. This narrowness of the synchronizability region, which +somewhat contrasts with the broader region obtained in frame of the 1D model +(Fig. 10 of Stefani, Giesecke and Weier (2019)), might have to do with the tight +scaling of Tn with the period of the meridional circulation. +4. Conclusions +As a sequel to the 0D and 1D modelling of solar cycle synchronization (Stefani +et al., 2016, 2018; Stefani, Giesecke and Weier, 2019; Stefani et al., 2020a; +Stefani, Stepanov and Weier, 2021), we have investigated a more realistic 2D +α − Ω-dynamo model. Starting from a conventional set-up without meridional +circulation, exhibiting a badly shaped butterfly diagram, via an enhanced model +with meridional circulation showing the correct butterfly shape, we have assessed +the synchronization capabilities of a time-periodic α term concentrated in the +tachocline region. For rather standard values of all other parameters, it was +shown that synchronization starts already for a magnitude of this additional +α-term as low as some dm/s. The smallness of this value relies on the fact that +η in the quiet tachocline region is significantly lower than in the convection +zone where it is dominated by the turbulent β effect. The utilized tachoclinic +diffusivity η ≈ 2.13 × 109 cm2/s should be considered a conservative choice; in +view of much lower values such as, e.g. 2.2 × 108 cm2/s as used by Guerrero and +de Gouveia Dal Pino (2007), the real value of α, required for synchronization, +might still be lower than the one derived here. +SOLA: klevs.tex; 16 January 2023; 1:27; p. 9 + +Be, r=0.95 +90 +latitude, [deg] +45 +0.25 +0 +0.00 +45 +-0.25 +-90 +Be, r=0.7 +90 +latitude, [deg] +10 +0 +0 +45 +90 +-10 +Br, r=0.95 +latitude, [deg] +90 +45 +0 +0 +45 +90 +Bl, r=0.95 +latitude, [deg] +90 +6 +0 +45 +2 +90 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +t, [yr]M. Klevs et al. +Figure 6. Ratio of the period Td of the signal to the period Tf of the forcing in dependence +on the relative strength of the forcing αp +max/αc +max. The color coded curves refer to different +ratios of the “natural” period Tn of the non-synchronized dynamo to Tf, which has been +varied by changing the magnetic Reynolds number Rm of the meridional circulation. Tn can +be read off from the value at the ordinate axis multiplied by 11 years; it amounts, for example, +to 23.3 years for Rm = 150, to 21.6 years for Rm = 200, and to 19.5 years for Rm = 250. +This brings us back to Charbonneau’s “elephant in the room: what, then, can +be considered a physically reasonable amplitude for external forcing?” (Charbon- +neau, 2022). Let us recall the very rough energetic consideration ¨Opik (1972) that +the typical tidal height of htidal = GmR2 +tacho/(gtachod3) ≈ 1 mm corresponds en- +ergetically to a velocity scale of v0 ∼ (2gtachohtidal)1/2 ≈ 1 m/s when employing +the huge gravity at the tachocline of gtacho ≈ 500 m/s2. Invoking the equally +rough estimate α ∼ v0 from renormalization theory (Moffatt and Dormy, 2019) +(and even when realistically assuming α to be one or two orders of magnitude +smaller than v0), a tidally generated α-value of a few dm/s seems not out of +reach. Indeed, it was recently shown (Horstmann, 2022) that (magneto-)Rossby +waves (Marquez-Artavia, Jones, and Tobias,, 2017; Zaqarashvili, 2018; Dikpati +et al., 2020) under the influence of a realistic tidal forcing are capable of acquiring +velocity scales of up to 1 m/s. Therefore, it appears that the “astrological home- +opathy” (Charbonneau, 2022) of tidal forcing may well be suited to generate an +α-effect in the tachocline region that is strong enough to entrain the entire solar +dynamo. +We have further confirmed the prior results of Stefani, Giesecke and Weier +(2019) (Figure 10) and Charbonneau (2022) (Figure 10) that this type of syn- +chronization requires a certain proximity of the tidal forcing’s period to half +the “natural” period of the undisturbed dynamo. The Sun, therefore, may just +be in the lucky situation of being orbited by a Jupiter with a period that fits +nicely to half the “natural” period of the undisturbed dynamo. It remains to +be seen whether some peculiar features of the solar dynamo, e.g its somewhat +unusual cycle period (B¨ohm-Vitense, 2007) and, in particular, “its comparatively +smooth, regular activity cycle” (Radick et al., 2018), could find an explanation +at this point. +SOLA: klevs.tex; 16 January 2023; 1:27; p. 10 + +Rm +2.2 +150.00 +160.00 +2.1 +170.00 +180.00 +190.00 +2.0 +200.00 +210.00 +1.9 +220.00 +230.00 +240.00 +1.8 +250.00 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ap +max +maxA synchronized two-dimensional α − Ω model of the solar dynamo +What are the next steps to be taken? First and foremost, the specific action +of m = 2 tidal forces on various m = 1 instabilities (e.g., Tayler) or waves (e.g., +magneto-Rossby), and on the α effect connected with them, has to be quantified +in a reliable manner. Complementary work on tidal influences on Rayleigh- +B´enard convection, and its large-scale circulation (Stepanov and Stefani, 2019; +J¨ustel et al., 2020, 2022), might be helpful to elucidate helicity entrainment in +a more generic sense. +Second, the possible role of further axisymmetric induction effects, beyond +the α effect, has to be clarified. The basic idea of a torque-influenced magnetic +buoyancy instability within the tachocline (Ferriz Mas, Schmitt, and Sch¨ussler, +1994; Zhang et al., 2003; Abreu et al., 2012) might play a central role here. It +was indeed employed as the basic synchronization mechanism by Charbonneau +(2022), while in Stefani et al. (2020a); Stefani, Stepanov and Weier (2021), we +had used it only to bring into play the second fundamental period 19.86 years +via spin-orbit coupling (yet poorly understood, but see Javaraiah (2003); Shirley +(2006); Sharp (2013) for first estimates). It certainly needs much more work to +disentangle these two effects. Further to this, we should not overlook alternative +axisymmetric (m = 0) instabilities, the possible relevance of which had been +discussed by several authors (Dikpati et al., 2009; Rogers, 2011). The recently +discovered helical magnetorotational instability for flows with positive radial +shear (Mamatsashvili et al., 2019) might be an particularly interesting candidate +in this respect. +Acknowledgments +This work received funding from the European Research Council (ERC) +under the European Union’s Horizon 2020 research and innovation programme (grant agree- +ment No 787544). We are grateful to Detlef Elstner for providing us with the finite-difference +dynamo code. F.S. would like to thank J¨urg Beer, Robert Cameron, Antonio Ferriz Mas, +Peter Frick, Gerrit Horstmann, Henri-Claude Nataf, Rafael Rebolo, G¨unther R¨udiger, Dmitry +Sokoloff, Willie Soon, Steve Tobias, Rodion Stepanov, Tom Weier, Ian Wilson and Teimuraz +Zaqarashvili for helpful discussions on various aspects of the solar dynamo and its possible +synchronization. L.J. acknowledges support from the Institut Universitaire de France. +Disclosure of Potential Conflicts of Interest +The authors declare that they have no conflicts of interest. +References +Abreu, J.A., Beer, J., Ferriz-Mas, A., McCracken, K.G., Steinhilber, F.: 2012, Is there a +planetary influence on solar activity? Astron. Astrophys. 548, A88. DOI. +B¨ohm-Vitense, E.: 2007, Chromospheric activity in G and K main-sequence stars, and what +it tells us about stellar synamos. Astrophys. J. 657, 486. DOI. +Bollinger, C.J.: 1952, A 44.77 year Jupiter–Venus–Earth configuration Sun-tide period in solar- +climatic cycles. Proc. 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J. 596, 663 DOI. +SOLA: klevs.tex; 16 January 2023; 1:27; p. 14 + diff --git a/pNE5T4oBgHgl3EQfIw6b/content/tmp_files/load_file.txt b/pNE5T4oBgHgl3EQfIw6b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b65ac36b1e37ef4b2233fb29489470c6b1f39c72 --- /dev/null +++ b/pNE5T4oBgHgl3EQfIw6b/content/tmp_files/load_file.txt @@ -0,0 +1,1060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf,len=1059 +page_content='Solar Physics DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='1007/•••••-•••-•••-••••-• A synchronized two-dimensional α − Ω model of the solar dynamo M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Klevs1,2 · F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani1 · L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Jouve3 © Springer •••• Abstract We consider a conventional α − Ω-dynamo model with meridional circulation that exhibits typical features of the solar dynamo, including a Hale cycle period of around 20 years and a reasonable shape of the butterfly diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' With regard to recent ideas of a tidal synchronization of the solar cycle, we complement this model by an additional time-periodic α-term that is localized in the tachocline region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' It is shown that amplitudes of some dm/s are sufficient for this α-term to become capable of entraining the underlying dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' We argue that such amplitudes of α may indeed be realistic, since velocities in the range of m/s are reachable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', for tidally excited magneto-Rossby waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Keywords: Solar cycle, Models Helicity, Theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Introduction The general idea that solar activity variations might be linked to the orbital motion of the planets traces back to Wolf (1859), and was kept alive, through- out one and a half centuries, by a number of authors (de la Rue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 1872;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Bollinger, 1952;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Jose, 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Takahashi, 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Wood, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' ¨Opik, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Condon and Schmidt, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Charvatova, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Zaqarashvili, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Landscheidt, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Palus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' De Jager and Versteegh, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Wolff and Patrone, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Abreu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Callebaut, de Jager, and Duhau, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The more specific coincidence, though, of the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='07-year alignment cycle of the tidally dominant planets Venus, Earth and Jupiter with the Schwabe cycle was brought to the � F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='stefani@hzdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='de 1 Helmholtz-Zentrum Dresden – Rossendorf, Bautzner Landstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 400, D-01328 Dresden, Germany 2 University of Latvia, Institute for Numerical Modelling, 3 Jelgavas street, Riga, LV-1004, Latvia 3 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Toulouse, IRAP, CNRS, UMR 5277,CNES,UPS, F-31400 Toulouse, France SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='05452v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='SR] 13 Jan 2023 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Klevs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' fore only recently by Hung (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Scafetta (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Wilson (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Okhlopkov (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Since even such a remarkable agreement between the average values of two periods might still be a pure coincidence, the question of whether there is a phase coherence between the two time series becomes of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The possible phase stability of the Schwabe cycle was first discussed in the paper “Is there a chronometer hidden deep in the Sun?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' by Dicke (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Analyzing the ratio between the mean square of the residuals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', the distances between the instants of the actual cycle maxima and the hypothetical maxima according to a linear trend) to the mean square of the differences between two consec- utive residuals, Dicke’s conclusions favoured a clocked process over a random walk process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' However, apart from the poor statistics connected with the mere 25 maxima taken into account, one should also take seriously Hoyng’s later warning (Hoyng, 1996) that any α-quenching mechanism could easily lead to a sort of self-stabilization of the solar dynamo, making a genuine random walk process “disguise” itself as a clocked process - at least for some centuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' A complementary type of cycle stability appears as a typical feature of conven- tional Babcock-Leighton dynamos whose period is largely determined by the turnover time of the meridional circulation (Dikpati and Charbonneau, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Charbonneau and Dikpati, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Charbonneau, 2020), which is indeed assumed to be much less fluctuating than the α effect in the convection zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' With those caveats in mind, we had recently re-considered (Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2020b) the longer time series of cycle minima/maxima as bequeathed to us by Schove (1983), and matched them with two series of the cosmogenic isotopes 10Be and 14C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Apart from the hardly decidable existence, or not, of two “lost cycles” (or phase jumps) around 1563 (Link, 1978) and 1795 (Usoskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2002), our analysis confirmed, by and large, Dicke’s conclusion in favour of a clocked cycle, now throughout the last millennium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' This result was then put into the context of the most remarkable, though widely overlooked, work of Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2004) whose analysis of two series of algae-related data from 10000-9000 cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' BP had evidenced a phase-stable Schwabe cycle with a period of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='04 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' In view of those two independent thousand-year long segments showing nearly identical Schwabe cycles with average periods between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='04 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='07 years and the strong evidence for phase stability in either case, we consider it at least worthwhile to quest for a possible physical mechanism that could be capable of linking the weak tidal forces as exerted by planets with the solar dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Setting out from the numerical observation (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2016) that a tide-like influence (with its typical m = 2 azimuthal dependence) can entrain the helicity oscillation of an underlying m = 1 instability (the Tayler instability (Tayler, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Seilmayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2012) for that matter) with barely changing its energy content, we have pursued some rudimentary synchronization studies in the framework of simple 0D and 1D α − Ω-dynamo models (Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Giesecke and Weier, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Within the same frame- work, we recently tried (Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Stepanov and Weier, 2021) to explain also the longer term Suess-de Vries cycle in terms of a beat period (Wilson, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Solheim, 2013) between the fundamental 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='14-year Hale cycle and the 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='86-yr period of the Sun’s barycentric motion (forced, in turn, by the SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 2 A synchronized two-dimensional α − Ω model of the solar dynamo orbits of Jupiter and Saturn (Cionco and Pavlov, 2018)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' With the intervening spin-orbit coupling remaining poorly understood, we took resort to the same buoyancy instability mechanism as it had been been employed by Abreu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2012) to plausibilize typical modulation periods on the centennial time-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Yet, this similarity between the final results notwithstanding, the fundamental time-scales of our model (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='14 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='86 years) that generate the much longer beat period of 193 years, are still close to the period of the undisturbed dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Our mechanism for explaining long-term modulations might, therefore, be less vulnerable to stochastic noise than what was discussed by Charbonneau (2022) in relation to the original model of Abreu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Admittedly, being restricted to the latitudinal coordinate, our simple 1D dynamo model did not have the requisite level of detail to give a quantitative answer to Charbonneau’s recent question of “what, then, can be considered a physically reasonable amplitude for external forcing” (Charbonneau, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' It was all the more encouraging that, utilizing a 2D Babcock-Leighton model with a periodic perturbation of the lower operating field threshold of the source term, Charbonneau (2022) found a similarly robust synchronization mechanism as Stefani, Giesecke and Weier (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Such a variation of the lower operating field threshold would correspond to variations of the field loss term κ as employed in (Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Stepanov and Weier, 2021) to parameterize the spin-orbit coupling with its 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='86-yr periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' While we do not exclude a viable physical translation of the (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='07-yr periodic) tidal forcing into such a type of variation of the field storage capacity, in this paper we will stick to our original idea that it is essentially the α effect that is affected by the tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Specifically, we seek to know then how much of this periodic α variation would be needed to accomplish synchronization of an otherwise conventional α−Ω-dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Guided by a rough estimation based on the virial assumption Upot ≈ Ekin, we consider approximately 1 m/s an upper limit for the tide-induced velocity variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Given that the value of α, which reflects only the helical part of the turbulence, is typically one order of magnitude lower than the underlying velocity, the focus of our modelling will be on whether α-values of the order of dm/s are sufficient to entrain the entire solar dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' To answer this very specific question, we step back from the more sophisti- cated double-synchronization model of Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Stepanov and Weier (2021) and restrict ourselves to the very basic tidal synchronization of the Schwabe/Hale cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' In the next section, we present a rather conven- tional two-dimensional α − Ω-dynamo with meridional circulation up, utilizing observation-constrained values for Ω and up, and employing more or less realistic values of α and the magnetic diffusivity η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' To keep the model simple, no specific Babcock-Leighton source term is added to the α-effect “living” in the convection zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' In the next section, we first adjust the value of η to provide a reasonable natural period of the undisturbed dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' While the most simple form of the α − Ω model leads, as usual, to a badly shaped butterfly diagram, the correct butterfly shape is recovered by switching on the meridional circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Based on the reference model thus defined, we will then assess in detail how much α variation in the tachocline region is actually needed for synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The paper will conclude with a short discussion of the results and some prospects for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Klevs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The model In this section, we motivate and describe our mean-field solar dynamo model and discuss its numerical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Considering only axi-symmetric solutions, we work with a system of partial differential equations whose spatial variables are the co-latitude and the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Intentionally, the model has been kept similarly simple as the benchmark model of Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' As usual, the magnetic field is split into a poloidal component BP (r, Θ, t) = ∇ × (A(r, Θ, t)eφ) and a toroidal component BT (r, Θ, t) = B(r, Θ, t)eφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The main sources of dynamo action are the gradient of the angular velocity Ω and the α-effect resulting from the helical part of the turbulence in the convection zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' While our model is not a Babcock-Leighton model (which would require a particular source term at the surface) it is a flux-transport model in that it comprises a meridional circulation up, mainly to ensure a realistic shape of the butterfly diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Choosing the solar radius R⊙ = 695700 km as the length and the diffusive time R2 ⊙/ηt as the time scale, we employ here - as in Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2008) - the dimensionless form of the coupled induction equations for the azimuthal components B := Bφ of the magnetic field and A := Aφ of the vector potential, ∂B ∂t = ˜ηD2B + 1 s ∂(sB) ∂r ∂˜η ∂r − Rmsup · ∇ �B s � + CΩs(∇ × (Aeφ)) · ∇Ω (1) ∂A ∂t = ˜ηD2A − Rm s up · ∇(sA) + Cc ααcB + Cp ααpB, (2) wherein we use the notations D2 := (∇2 − s−2), s := r sin θ and ˜η = η/ηt, with ηt being the turbulent magnetic diffusivity in the convection zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' This system is governed by four magnetic Reynolds numbers characterizing, respectively, the effects of shear, meridional circulation, and two different α terms: CΩ = ΩeqR2 ⊙/ηt (3) Rm = u0R⊙/ηt (4) Cc α = αc maxR⊙/ηt (5) Cp α = αp maxR⊙/ηt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (6) Herein, Ωeq = 2π × 456 nHz is the angular velocity at the equator, and u0 and αc max and αp max are the typical intensities of the meridional circulation and the two separate α effects in the convection zone and in the tachocline region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' In con- trast to Guerrero and de Gouveia Dal Pino (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2014), we do not incorporate any specific Babcock-Leighton source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' We suppose the turbulent magnetic diffusivity ηt in the convection zone to be dominated by a strong β effect, whereas it is much smaller in the relatively quiet tachocline region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Refraining from more complicated structures of η as employed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', in Guerrero and de Gouveia Dal Pino (2007) or Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 4 A synchronized two-dimensional α − Ω model of the solar dynamo Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Spatial structures of the main ingredients of the dynamo model in the meridional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (a) Isolines of Ω(r, Θ)/Ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (b) Streamlines of up(r, Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (c) Constant part of α, taken in the unquenched state: αc(r, Θ)/αc max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (d) Periodic part of α, with the resonance term set to 1: αp(r, Θ)/αp max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2014), we use here the simple form of Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2008) ˜η(r) = ηc ηt + 1 2 � 1 − ηc ηt � � 1 + erf �r − rc d �� (7) with ηc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='01ηt, rc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='02, which shows a smoothed-out jump (by a factor of 100) between the radiation zone and the convection zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' For the angular velocity we apply the same spatial structure as in Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2008): Ω(r, Θ) = CΩ � Ωc + 1 2 � 1 + erf �r − rc d �� (1 − Ωc − c2 cos2 Θ) � (8) with rc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='02, Ωc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='92 and c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 (see Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' For the meridional circulation we chose, again as in Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2008), one single cell defined by up = ∇ × (ψ(r, Θ)eφ) with the stream function ψ(r, Θ) = Rm � − 2 π (r − rb)2 (1 − rb) sin � π r − rb 1 − rb � cos Θ sin Θ � (9) with rb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 (see Figure 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' We are well aware of the fact that the specific structure of up is much less settled than that of Ω(r, Θ), and that more compli- cated two-cell flows (Kosovichev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2022) might also be considered in future improvements of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Finally, α = αc + αp is thought to consist of a conventional part αc in the convection zone, whose time-dependence stems only from the quenching by the magnetic field, αc(r, Θ, t) = Cc α 3 √ 3 4 sin2 Θ cos Θ � 1 + erf �r − rc d �� � 1 + |B(r, Θ, t)|2 B2 0 �−1 (10) SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 5 90° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='06 90° 45° 45° 45° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 0° 0° 0° 0° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 45° 45° 45° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 90° 90° 90° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='06- (a) (b) (c) (d)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Klevs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' with B0 = 1, and an explicitly time-dependent (with forcing period Tf) part αp that is concentrated in the tachocline region, αp(r, Θ, t) = Cp α 1 √ 2 sin2 Θ cos Θ � 1 + erf �r − rc d �� � 1 − erf �r − rd d �� × × |B(r, Θ, t)|2 1 + |B(r, Θ, t)|4 sin(2πt/Tf) , (11) where rd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Note that the factor on the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (11) represents a resonance term as introduced in Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2016) in order to account for a field-dependent optimal reaction of the underlying instability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', Tayler instability) on the tidal forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' A similar field dependence has been used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', in Charbonneau (2022), although with the slightly different interpretation as a nonlinearity of the non-local source term that incorporates both a lower and upper operating threshold on the strength of the toroidal magnetic at the base of the convection zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The spatial structures of these two α terms are visualized in Figure 1(c,d), in either case disregarding any magnetic-field dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' For the numerical solution, an explicit finite difference scheme in two di- mensions in spherical coordinates is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' As in R¨udiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2003), the standard resolution was 64 × 64 grid points in both radial and latitudinal di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The equations are solved with perfect conductor boundary conditions A = ∂(rB)/∂r = 0 at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65R⊙ and vertical field conditions Bφ = BΘ = 0 at r = R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Results In this section we present and assess the results of three dynamo models with increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Non-synchronized model, without meridional circulation First we consider the simplest case of a Parker’s migratory dynamo (Parker, 1955), without any synchronization term (αp = 0), and without meridional circulation (up = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' For the sake of concreteness, we set ηt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='13×1011 cm2/s, and αc max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='30 m/s, which both are close to the respective geometric means of the lower and upper values as typically found in the literature (1010 −1013 cm2/s for η and 10 − 103 cm/s for α, see Charbonneau (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The resulting magnetic Reynolds numbers according to Equations (3) and (5) are CΩ = 65100 and Cc α = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The radial dependencies of η(r) and αc (in its unquenched form) are illustrated, for Θ = 45◦, in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Note that at this particular angle αc(r) does not reach the maximum value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='30 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Figure 3 illustrates the resulting field dependence on time and latitude, taken at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95, showing a reasonable dynamo cycle period of Td = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='27 years (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='0198 diffusion times), but a badly shaped butterfly diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 6 A synchronized two-dimensional α − Ω model of the solar dynamo Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Radial dependence of various dynamo ingredients in physical units, all taken at Θ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (a) Diffusivity η(r) (black), αc(r) in the unquenched form (violet), and αp(r) for αp max = αc max and with the field-dependent resonance factor artificially set to 1 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (b) uΘ(r) resulting from the stream function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (9) for three different Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Contour-plots BΘ(r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95, θ, t), Bφ(r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7, θ, t), and Br(r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95, θ, t) and of |B(r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95, θ, t)| for the non-synchronized model without meridional circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Note that the ordinate axis represents not the colatitude θ but the normal solar latitude 90◦ − θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Non-synchronized model, with meridional circulation In order to recover the correct shape of the butterfly diagram, we switch on a meridional circulation, setting its value to u0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 m/s, which corresponds to Rm = 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' For this value, as well as for Rm = 200 and 240, the radial dependence of uΘ is shown, again for Θ = 45◦, in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' While the values uΘ at r = 1 are by factor of appr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' two too low compared with observations, the typical values of 1-2 m/s at the base of the convection zone are quite compatible with values from helioseismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Actually, the latter velocities are the crucial ones to set the cycle period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' As seen in Figure 4 we obtain now a butterfly diagram of rather decent shape, and a slightly changed cycle period of Td = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='798 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' This will serve in the fol- SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 7 (a) (b) 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='4 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='6 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='4 [s/w] S 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 [m/ 3 Q 1 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='8 4 uα for Rm=170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='6 5 Qb Rm=200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='4 6 Rm=240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 7 0 0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 90 latitude, [deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 90 B, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 latitude, [deg] 90 10 45 0 0 45 10 90 Br, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 90 latitude, [deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='0 0 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 90 90 latitude, [deg] 45 3 0 45 90 0 0 25 50 75 100 125 150 175 200 t, [yr]M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Klevs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Same as Figure 3, but including meridional circulation with Rm = 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' lowing as the reference dynamo model whose synchronization is to be evaluated thereupon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' While further improvements of the spatio-temporal features of the magnetic field are certainly possible (for example, when including an appropriate Babcock-Leighton source term), we refrain from any further sophistication of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Synchronized model Finally, we switch on the periodic α term with an assumed forcing period of Tf = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 years (we do not insist here on the precise value of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='07 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The radial dependence of αp is illustrated by the red curve in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Note, however, that here αp max has the same value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='30 m/s as the corresponding αc max, and that the field-dependent resonance term in Equation (11) is artificially set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' In reality, the resonance term reduces this value by a factor of 2 for the optimum field strength, and even more so outside the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' As shown in Figure 5, for the specific value αp max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='52 m/s we obtain now the dynamo period Td = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 years which corresponds to twice the period Tf of the forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' In Figure 6 we plot the dependence of the dynamo period Td on αp max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Here we have used a couple of ratios of the “natural” period Tn (of the non-synchronized dynamo with αp max = 0) to the forcing periods Tf by simply changing the am- plitude of meridional circulation which governs Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Very similar to Figure 10 in Stefani, Giesecke and Weier (2019), and to Figure 10 in Charbonneau (2022), we obtain a clear parametric resonance for some critical value of αp max that depends on the initial distance between twice the forcing period Tf and the natural period Tn of the unperturbed dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' As we had chosen αc max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='30 m/s, synchronization occurs for an amplitude of αp max in the range of some dm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The relative smallness of this number is, of course, a consequence of the 100 times smaller value of η in the tachocline region which amplifies correspondingly the SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 8 Be, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 90 latitude, [deg] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 90 Bs, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 90 latitude, [deg] 10 45 0 0 45 10 90 Br, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 latitude, [deg] 90 45 0 0 45 90 Bl, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 latitude, [deg] 90 6 4 45 2 90 0 0 25 50 75 100 125 150 175 200 t, [yr]A synchronized two-dimensional α − Ω model of the solar dynamo Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Same as Figure 4, but with synchronization by a periodic α-term with amplitude αp max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='52 m/s and period Tf = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' induction effect of αp, even if the latter is concentrated in a significantly smaller zone than αc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' That said, we must also admit that synchronization requires a certain proximity of 2Tf and Tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' for the Rm values indicated by the dashed lines in Figure 6 no clear synchronization was observed even for the highest considered value of αp max/αc max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' This narrowness of the synchronizability region, which somewhat contrasts with the broader region obtained in frame of the 1D model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 10 of Stefani, Giesecke and Weier (2019)), might have to do with the tight scaling of Tn with the period of the meridional circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Conclusions As a sequel to the 0D and 1D modelling of solar cycle synchronization (Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Giesecke and Weier, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Stepanov and Weier, 2021), we have investigated a more realistic 2D α − Ω-dynamo model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Starting from a conventional set-up without meridional circulation, exhibiting a badly shaped butterfly diagram, via an enhanced model with meridional circulation showing the correct butterfly shape, we have assessed the synchronization capabilities of a time-periodic α term concentrated in the tachocline region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' For rather standard values of all other parameters, it was shown that synchronization starts already for a magnitude of this additional α-term as low as some dm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The smallness of this value relies on the fact that η in the quiet tachocline region is significantly lower than in the convection zone where it is dominated by the turbulent β effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The utilized tachoclinic diffusivity η ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='13 × 109 cm2/s should be considered a conservative choice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' in view of much lower values such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 × 108 cm2/s as used by Guerrero and de Gouveia Dal Pino (2007), the real value of α, required for synchronization, might still be lower than the one derived here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 9 Be, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 90 latitude, [deg] 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='25 90 Be, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 90 latitude, [deg] 10 0 0 45 90 10 Br, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 latitude, [deg] 90 45 0 0 45 90 Bl, r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='95 latitude, [deg] 90 6 0 45 2 90 0 0 25 50 75 100 125 150 175 200 t, [yr]M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Klevs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Ratio of the period Td of the signal to the period Tf of the forcing in dependence on the relative strength of the forcing αp max/αc max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The color coded curves refer to different ratios of the “natural” period Tn of the non-synchronized dynamo to Tf, which has been varied by changing the magnetic Reynolds number Rm of the meridional circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Tn can be read off from the value at the ordinate axis multiplied by 11 years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' it amounts, for example, to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='3 years for Rm = 150, to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='6 years for Rm = 200, and to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='5 years for Rm = 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' This brings us back to Charbonneau’s “elephant in the room: what, then, can be considered a physically reasonable amplitude for external forcing?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (Charbon- neau, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Let us recall the very rough energetic consideration ¨Opik (1972) that the typical tidal height of htidal = GmR2 tacho/(gtachod3) ≈ 1 mm corresponds en- ergetically to a velocity scale of v0 ∼ (2gtachohtidal)1/2 ≈ 1 m/s when employing the huge gravity at the tachocline of gtacho ≈ 500 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Invoking the equally rough estimate α ∼ v0 from renormalization theory (Moffatt and Dormy, 2019) (and even when realistically assuming α to be one or two orders of magnitude smaller than v0), a tidally generated α-value of a few dm/s seems not out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Indeed, it was recently shown (Horstmann, 2022) that (magneto-)Rossby waves (Marquez-Artavia, Jones, and Tobias,, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Zaqarashvili, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Dikpati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2020) under the influence of a realistic tidal forcing are capable of acquiring velocity scales of up to 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Therefore, it appears that the “astrological home- opathy” (Charbonneau, 2022) of tidal forcing may well be suited to generate an α-effect in the tachocline region that is strong enough to entrain the entire solar dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' We have further confirmed the prior results of Stefani, Giesecke and Weier (2019) (Figure 10) and Charbonneau (2022) (Figure 10) that this type of syn- chronization requires a certain proximity of the tidal forcing’s period to half the “natural” period of the undisturbed dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The Sun, therefore, may just be in the lucky situation of being orbited by a Jupiter with a period that fits nicely to half the “natural” period of the undisturbed dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' It remains to be seen whether some peculiar features of the solar dynamo, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g its somewhat unusual cycle period (B¨ohm-Vitense, 2007) and, in particular, “its comparatively smooth, regular activity cycle” (Radick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2018), could find an explanation at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' SOLA: klevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 16 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 1:27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' 10 Rm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='1 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='9 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='8 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='8 ap max maxA synchronized two-dimensional α − Ω model of the solar dynamo What are the next steps to be taken?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' First and foremost, the specific action of m = 2 tidal forces on various m = 1 instabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', Tayler) or waves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', magneto-Rossby), and on the α effect connected with them, has to be quantified in a reliable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Complementary work on tidal influences on Rayleigh- B´enard convection, and its large-scale circulation (Stepanov and Stefani, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' J¨ustel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2020, 2022), might be helpful to elucidate helicity entrainment in a more generic sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Second, the possible role of further axisymmetric induction effects, beyond the α effect, has to be clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The basic idea of a torque-influenced magnetic buoyancy instability within the tachocline (Ferriz Mas, Schmitt, and Sch¨ussler, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Abreu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2012) might play a central role here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' It was indeed employed as the basic synchronization mechanism by Charbonneau (2022), while in Stefani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, Stepanov and Weier (2021), we had used it only to bring into play the second fundamental period 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='86 years via spin-orbit coupling (yet poorly understood, but see Javaraiah (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Shirley (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Sharp (2013) for first estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' It certainly needs much more work to disentangle these two effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Further to this, we should not overlook alternative axisymmetric (m = 0) instabilities, the possible relevance of which had been discussed by several authors (Dikpati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Rogers, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' The recently discovered helical magnetorotational instability for flows with positive radial shear (Mamatsashvili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 2019) might be an particularly interesting candidate in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Acknowledgments This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agree- ment No 787544).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' We are grateful to Detlef Elstner for providing us with the finite-difference dynamo code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' would like to thank J¨urg Beer, Robert Cameron, Antonio Ferriz Mas, Peter Frick, Gerrit Horstmann, Henri-Claude Nataf, Rafael Rebolo, G¨unther R¨udiger, Dmitry Sokoloff, Willie Soon, Steve Tobias, Rodion Stepanov, Tom Weier, Ian Wilson and Teimuraz Zaqarashvili for helpful discussions on various aspects of the solar dynamo and its possible synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=': 2017, The Tayler instability at low magnetic Prandtl numbers: chiral symmetry breaking and synchronizable helicity oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Magnetohydrodynamics 53, 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=' Stefani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', Giesecke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', Weber, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE5T4oBgHgl3EQfIw6b/content/2301.05452v1.pdf'} +page_content=', 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polytopes from the perspective +of both n-dimensional geometry and abstract symmetry groups. It starts +with a brief introduction to the terminology and a short motivation. In +the first part, it engages in the construction of all regular tessellations and +polytopes of n dimensions and extends this to the study of their quasi- +regular and uniform generalizations. In the second part, the symmetries +of polytopes and tessellations are considered and the Coxeter groups and +their associated root systems are introduced and classified. In the last +part, the algorithms developed for this work are described and their results +discussed. +1 + +Contents +1 +Introduction +3 +1.1 +Motivation and tessellation definitions . . . . . . . . . . . . . . . +3 +1.2 +Polytope definitions +. . . . . . . . . . . . . . . . . . . . . . . . . +4 +2 +The regular tessellations and polytopes of n dimensions +5 +2.1 +The regular tilings and polyhedra . . . . . . . . . . . . . . . . . . +5 +2.2 +The regular cubic honeycomb and polychora . . . . . . . . . . . . +8 +2.3 +The regular tessellations in n-1 dimensions and n-polytopes . . . +11 +3 +The quasi-regular and uniform tessellations and polytopes +14 +3.1 +Quasi-regular tilings and the crystallographic restriction . . . . . +14 +3.2 +Uniform honeycombs and geometric operations . . . . . . . . . . +16 +4 +Coxeter groups and reflection groups +18 +4.1 +Coxeter systems +. . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4.2 +Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4.3 +Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +5 +The hypercube tessellation and convex polytope algorithms +24 +5.1 +The developed algorithms . . . . . . . . . . . . . . . . . . . . . . +24 +5.2 +Obtained results +. . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +Appendices +29 +A Visualizations +29 +Bibliography +30 +2 + +1 +Introduction +1.1 +Motivation and tessellation definitions +The idea for this work started from the visualization of the sequence of 4-cube +cross-sections with a 3-flat (3-dimensional affine space) which results in an oc- +tahedron at the center of the 4-cube and in tetrahedrons of different size and +truncation magnitude along any of the 4-fold diagonals of the 4-cube. +Figure 1: 3-dimensional cross-sections along 4-fold symmetry axis. +The fact that the three regular polytopes in all dimensions could be obtained +from a cross-section with a hypercube and that various uniform polytopes can +further be obtained from these together with the fact that the hypercubic tes- +sellation is the only regular tessellation in all dimensions led to the idea to +develop an algorithm that could help us identify specific types of symmetric +cross-sections with the hypercubic tessellation. +Since there were many different naming conventions used by mathematicians re- +garding some of the terms studied in this text, this first subsection is devoted to +a clarification regarding those. Most mathematicians distinguish among tilings, +honeycombs and tessellations, using the last as a generalization of n-dimensional +case and the first two to thus refer to tessellations on the plane and on a higher +dimensional space. Here we employ the same meaning for these terms but avoid +their interchangeability which is possible in some sources. +The definition of a tessellation used here is due to Coxeter [3]. A tessellation is +an infinite set of polytopes fitting together to fill an n-dimensional space with- +out overlapping so that the n-1-facet of each polytope belongs to one and only +one other polytope. +Tessellations in n − 1 dimensions can be considered as n-dimensional apeiro- +topes, i.e. polytopes with infinitely many cells. This will be justified in the +next section where the connection between polytopes and tessellations will be +examined and it will be shown that a polytope may also be regarded as a tes- +sellation of a given manifold. Before this, a few definitions regarding polytopes +3 + +are necessary. +1.2 +Polytope definitions +A polytope P is a geometrical figure bounded by finitely many hyperplanes. +It is the n-dimensional generalization of polygon (2-polytope), polyhedron (3- +polytope), polychoron (4-polytope), etc. with the following properties: +• The facets of P are polytopes. +• P has facets of every dimension 0, 1, ..., n−1 and for brevity a k-dimensional +facet is called a k-facet of P. +• Every facet of a facet of P is also a facet of P. +• A convention assumed in this text is the existence of one n-facet of P +which is P itself. +• For every two facets F1 and F2 of P, F1 ∩ F2 is also a facet of P. +• For every two facets F1 and F2 of P there exists a uniquely defined facet +F1∨F2 of P, namely the smallest facet in terms of inclusion which contains +both F1 and F2 . +The last two properties originate from the following observation. +For an n- +polytope, each n-1-facet lies in a bounding hyperplane. Two or more adjacent +bounding hyperplanes intersect at n-2-flat and n-2-facet of P lies in this n-2-flat, +three or more adjacent n-2-flats intersect at n-3-flat and n-3-facet lies in this +n-3-flat and so on as n or more 1-flats or lines intersect in points which are the +vertices of the polytope. +There are different naming conventions of the various facets of a polytope besides +the obvious for vertex, edge, and face. Examples are cell for 3-facet, hypercell +for 4-facet, facet for n-1-facet, ridge for n-2-facet, and peak for n-3-facet. For +the sake of any further generalization, only the first part of these will be used. +Each constituent which is also a polytope of a lower dimension is additionally +referred to as k-face, k-boundary or similar alternative names but here k-facet +(k < n) will be used for this purpose. +Let Nr denote the number of r-facets of a polytope and Npq denote the number +of p-facets that lie in a q-facet if p < q or the number of p-facets that pass +through a q-facet if p > q. Then Nr = Nrr = Nrn and it can be noticed that +these numbers are not independent, i.e. NpqNq = NqpNp. They will be called +configurational numbers and will be used later in this work. +Polytopes may be divided into: convex and non-convex (also known as star- +polytopes); regular, quasi-regular and semi-regular; uniform and non-uniform. +4 + +A set S ⊂ Rn is convex if it has the property that for any pair of points x, y ∈ S, +the line segment +λx + (1 − λ)y : 0 ≤ λ ≤ 1 +with end points x and y, lies entirely in S. For any set S, the smallest convex +set containing S (the intersection of the family of all convex sets that contain +S) is called the convex hull of S, and is denoted by conv(S). A convex polytope +P is defined to be the convex hull of any finite set of points in Rn [1]. If F 0 +1 , ..., +F 0 +r are the end-points of the edges of P that meet at F 0, then conv(F 0 +1 , ..., F 0 +r ) +is an n-1-polytope called the vertex figure of P. An n-polytope P is regular if +its facets are regular and its vertex figures are regular (this definition will be +revisited in the next section). Finally, an n-polytope P is uniform if its facets +are uniform and its vertex figures are all of the same kind (P is vertex-transitive). +The dual of a polytope is defined as a polytope whose r-facets correspond to the +(n − r − 1)-facets of the original so that its n-1-facets are the vertex figures of +the original. The configurational numbers of the n-1-facet of the dual polytope +then naturally correspond to the configurational numbers of the vertex figure of +the original polytope. +Star-polytopes will not be considered in this work. +2 +The regular tessellations and polytopes of n +dimensions +2.1 +The regular tilings and polyhedra +In the following section we will enumerate the regular polytopes in n dimen- +sions and will use the same principle to define all regular tessellations of n-1 +dimensions as infinite regular n-polytopes. The construction of these is based +on Sommerville [2]. +Regular polyhedra satisfy two conditions: (1) their faces are regular polygons +of the same kind; (2) their solid angles are equal. In the classical definition +however, the second condition is restated in a stronger form: the vertex figures +are regular polygons, which allows for the first condition to be weakened to: +their faces are regular polygons. A more modern definition requires an even +stronger condition - the polyhedron must be transitive on its flags, i.e. face-, +edge-, and vertex-transitive, i.e. all faces, edges and vertices must be the same. +The reason for this new definition is that while the requirement for regular faces +and vertex figures is sufficient to derive this condition in the usual case, it is no +longer valid for the newly introduced abstract polytopes. +The two final conditions to be used for the following construction are weaker: +(1) each face has the same number n of edges and vertices; (2) there is the same +5 + +number p of edges and faces around each vertex. These two numbers n and p will +show to be sufficient for the construction of the same final polytopes as with the +stronger conditions. From this definition then it follows that N02 = N12 = n +and N10 = N20 = p where Nij is the number of i-faces incident to a j-face. +Since each edge is surrounded by two vertices and faces, N01 = N21 = 2. There- +fore using NijNj = NjiNi we arrive at nN2 = 2N1 = pN0. We have to now +add one more restriction that will ultimately limit the possible values of n and p. +This restriction is Euler’s polyhedron formula +N2 − N1 + N0 = χ +where χ denotes a topological invariant called the Euler characteristic. A regular +polyhedron then has the Euler characteristic of the sphere χ = 2. To verify this, +we need some further tools. The Gauss - Bonnet formula +� +M +KdA = 2πχ(M) +relates the Gaussian curvature K to the Euler characteristic χ as M is any +orientable closed surface. Descartes’ law of closure defect then states that if the +polyhedron is homeomorphic to a sphere (hence not necessarily convex), the +total angle defect as the sum of the defects of all the vertices is 4π which is the +total Gaussian curvature of the sphere. The Gauss - Bonnet formula then gives +4π = +� +M +KdA = 2πχ(M) ⇒ χ(M) = 2. +The surface of a polyhedron can be considered as a limit case of differentiable +surface where the total Gaussian curvature K remains zero at the faces and +edges of the polyhedron and is therefore concentrated on the discrete vertex +points. Although K is not defined at these points, its integral remains finite +and the integral of the total curvature can be replaced with the sum of the +angle defect at all vertices. Thus, the number of vertices can be easily derived +by dividing 2πχ(M) = 4π with the angle defect at a vertex. +N0(2 + 2n +p − n)π = 4π +The options for the selection of p polygons around a vertex and n vertices around +a polygon to get an appropriate angle defect are finally limited to this divisibility +criterion. Although this provides with solutions for p and n, a more geometric +approach which can be applied to a higher dimensional case is considered here. +From Euler’s formula for a regular polyhedron N2 − N1 + N0 = 2 we calculate +N0 = 2−N2+N1 = 2− pN0 +n + pN0 +2 +⇒ N0(1+ p +n − p +2) = 2 ⇒ N0 = +4n +2(n + p) − np +6 + +and analogously for the rest we get +N1 = +2np +2(n + p) − np, N2 = +4p +2(n + p) − np, λ = +2 +2(n + p) − np +which simplifies to N0 = 2nλ, N1 = npλ, and N2 = 2pλ. +Restricting this to finite regular polyhedra requires finite and positive λ which +imposes the additional conditions p > 2, n > 2 and 2(n+p)−np > 0 ⇒ p < +2n +n−2. +Therefore, for n = 3 ⇒ 2 < p < 6, n = 4 ⇒ 2 < p < 4, n = 5 ⇒ 2 < p < 4. +n ≥ 6 would imply 2 < p < 3 which is not possible, therefore this depletes all +options. As a result, all possible regular polyhedrons are +n +3 +3 +3 +4 +5 +p +3 +4 +5 +3 +3 +These are exactly the Platonic solids. The tetrahedron {3,3} is self-dual, while +the hexahedron {4,3} has the octahedron {3,4} as its dual and the icosahedron +{3,5} is the dual of the dodecahedron {5,3}. In a similar way we can take an +infinite regular polyhedron by setting 2(n + p) − np = 0 for which λ becomes +infinite. Since p > 2 and n > 2, this leads to +n +3 +4 +6 +p +6 +4 +3 +These are the three regular tilings, respectively the triangular {3,3}, the hexag- +onal {6,3}, and the square {4,4}. +The case 2(n+p)−np < 0 is a third case but in order to understand it better, we +must interpret these results first. In the first case, the polyhedron’s configuration +of vertices, edges and faces is isomorphic to a tessellation of a sphere where all +its vertices lie on the surface and its edges are represented by geodesic arcs. To +see this, let S denote the total sum of angles of an n-gon on the sphere with +radius r. Since the total angle deficiency for a sphere is +α = +� +M +KdA = 1 +r2 A(M) ⇒ A(M) = r2α +and for an n-gon +2π − α + S = nπ ⇒ α = (S − (n − 2)π) +it follows that the area of the n-gon must be A(M) = (S −(n−2)π)r2 and since +�N2 +i=1 Si = �N2 +i=1 S = 2πN0, and nN2 = 2N1 = pN0, the total area must be +((2N0 − (n − 2)N2)πr2 = (2N0 + 2pN0 +n +− pN0)πr2 = 2(n + p) − np +n +N0πr2. +Then for the case 2(n + p) − np = 0 either r, N0 or both have to be infinite in +order to have a finite area of the n-gon. If r is infinite the sphere becomes a +7 + +plane else N0 must be infinite which makes all vertices indefinitely close to each +other on a finite sphere thus again making Euclidean geometry applicable (with +sum of angles of n-gon now S = (n − 2)π). The third case follows from the first +two. +The last condition 2(n + p) − np < 0 then implies r2 < 0 where a purely imagi- +nary radius would require trigonometric formulae which hold on a surface with +constant negative Gaussian curvature K called Lobachevski sphere or a hyper- +bolic plane. These three cases then help us conclude that regular polyhedrons +are equivalent to tessellations of elliptic, Euclidean and hyperbolic plane. +The above condition 2(n + p) − np > 0 can also be rewritten as +2(n + p) − np > 0 ⇒ 2n +p + (2 − n) > 0 ⇒ 2 +p > 1 − 2 +n ⇒ 1 +p + 1 +n > 1 +2 +which is another inequality related to the vertex figure’s angle defect. +2.2 +The regular cubic honeycomb and polychora +In order to continue this construction in n dimensions, we now use the more +restrictive classical definition for n-dimensional regular polytope with the two +conditions: (1) all n-1-facets of the polytope must be regular polytopes; (2) all +vertex figures of the polytope must be regular polytopes. Although it is possible +to consider polytopes with n-1-facets that are tessellations of n-2-dimensional +space, a restriction only to finite polytopes as n-1-facets and vertex figures is +considered for the purpose of this construction. +To continue the approach from the previous subsection, extend the configura- +tional numbers Npq of the polytope with configurational numbers of the n-1-facet +Fpq and of the vertex figure Vpq. Then the following equalities hold because of +the definition: +Npq = Fpq +(p < q < 3), +Np3 = Fp +(p = 0, 1, 2) +Npq = Vp−1,q−1 +(p > q > 0), +Np0 = Vp−1 +(p = 1, 2, 3) +The above equalities are true for the specific values of p. +The first equal- +ity constitute the relations between the facet and the vertex figure of a 4- +polytope. +The second is derived from Np3 = Fp3 = Fp since the n-1-facets +of the 3-polytope are 3-facets. +The third equality makes use of duals N ′: +Vpq = F ′ +pq = N ′ +pq = Np−1,q−1. Finally, the last equality is similar to the second +in terms of the dual polytope. +Furthermore, the number of lines or planes through a point F10, F20 in a 3-facet +(cell) must be the same as the number of lines or planes V02, V12 through each +vertex, hence F10 = F20 = V02 = V12. This is one of the three numbers p, q, r +which can finally be extracted from +8 + +F02 = F12 = N02 = N12 = p +F10 = F20 = V02 = V12 = q +V10 = V20 = N21 = N31 = r +where p, q describe the 3-facet and q, r the vertex figure. They represent the +number of vertices or edges of a polygon (p), the number of edges or planes +through each vertex of a polyhedron (q), and the number of 3-facets or cells +through each edge (r). +As mentioned before, by superimposing all available choices for finite polyhe- +drons, the following 4-dimensional cases can be constructed: +333 +334 +335 +343 +353 +433 +434 +435 +533 +534 +535 +However, they still need to be distinguished in terms of metric (elliptic, Eu- +clidean, and hyperbolic). To do this, we have to examine the dihedral angles of +the constituent polyhedrons. The geometric prescription makes use of spherical +trigonometry. +Figure 2: Dihedral angles of a polyhedron. +Let C0 be any vertex a regular polyhedron p,q. Let C1 be the mid-point of an +edge through C0, C2 the center of a face though C0C1, and C3 the center of the +polyhedron. Denote ∠C1C0C2 = θ1 and ∠C2C1C3 = θ2. Since the polyhedron +is regular ∠C0C2C1 = π/p. Consider the unit sphere around C0 - it is cut by +the edges and faces through C0 into spherical polygon with q sides (the vertex +figure of the polyhedron). Then θ1 is half the length of the edge of any such +polygon and θ2 is half the magnitude of its angle. Finally, since π/q is half the +angle which an edge of the polygon subtends at its center, we can use Napier’s +rules for a spherical right triangle +9 + +C1 +3b=, +B=元/qcos(B) = sin(A)cos(b) → cosπ +q = sinθ2cosθ1 = sinθ2sinπ +p +where 2θ2 is the dihedral angle of the polyhedron. +The results for the dihedral angles of the regular polyhedrons with a given p +and q determine how many polyhedrons can be placed around an edge of the +constructed polychoron (4-polytope) which is the value of r. They are shown in +the next table: +p +q +2θ2 +r +3 +3 +70.5o +3, 4, 5 +3 +4 +109.5o +3 +4 +3 +90o +3, 4 +3 +5 +138.2o +- +5 +3 +116.6o +3 +Thus, the only case of polyhedra completely filling a space is the case of 4 +cubes around an edge (8 around a vertex) which identifies the case 434 as the +only 4-polytope tessellating an Euclidean 3-dimensional space, i.e. the cubic +honeycomb which is the only regular honeycomb in 3 dimensions. The regular +closed polychora which can be viewed as regular tessellations of a hypersphere +can only have 3, 4 or 5 tetrahedra, 3 octahedra, 3 or 4 hexahedra (cubes) or 3 +dodecahedra at an edge. They represent the cases 333, 334, 335, 343, 433, and +533 where 333 and 343 are self-dual, and the others are in dual pairs as 433 and +334 and 335 and 533. +Using the two conditions for regularity, the following relations can be obtained +N0qrq = N1, N1r = N2p, N2 = N3qqp +where +qp = +2 +2(p + q) − pq , rq = +2 +2(q − r) − qr. +However, they will not be explained further because Euler’s equation N0 −N1 + +N2 − N3 = 0 is homogeneous in four dimensions and can only determine the +ratios between these configurational numbers. +N0 : N1 : N2 : N3 = pqp : pqqprq : qrpqrq : rrq +Therefore, in order to finally determine these polychora, we would have to build +them geometrically. Since this gets beyond the scope of this work, here they +will simply be identified for the reader: +• {3,3,3} pentachoron, 5-cell, or 4-simplex (analog of the tetrahedron) +• {4,3,3} octachoron, 8-cell, or 4-cube (analog of the cube) +10 + +• {3,3,4} hexadecachoron, 16-cell, or 4-orthoplex (analog of the octahedron) +• {3,4,3} icositetrachoron, 24-cell, or octaplex (no 3-dimensional analog) +• {3,3,5} hexacosichoron or 600-cell (analog of the icosahedron) +• {5,3,3} hecatonicosachoron or 120-cell (analog of dodecahedron) +One of the algorithms developed for this thesis generates the coordinates of any +of these 4-dimensional analogues of the Platonic solids so that various cross- +sections can be performed in order to study their symmetry. +2.3 +The regular tessellations in n-1 dimensions and n- +polytopes +As we have seen, the problem of constructing a regular n-polytope is only a part +of the general problem of constructing a regular tessellation in n-1 dimensions. +Such a tessellation divides n-1-dimensional space into equivalent n-1-polytopes +which in turn are tessellations of elliptic space in n-2 dimensions. In addition, +any hypersphere around a vertex of the n-1-dimensional tessellation will define +a tessellation of elliptic space in n-2 dimensions. +Figure 3: Front view of a cube with cases p = 2, q = 0 (Aq-vertex, Ap+1-cube) +and p = 2, q = 1 (Aq-edge, Ap+1 − cube). +Consider a p+1-facet Ap+1 and a q-facet Aq inside it such that q ≤ p. In Ap+1 +take any (p + 1 − q)-dimensional flat Sp+1−q intersecting Aq at a point O and +construct a small hypersphere centered at O. It is easy to see that +dimSp+1−q = p + 1 − q = dimAp+1 − dimAq, +dimAq ∩ Sp+1−q ≤ min(q, p + 1 − q), and O ⊂ Aq ∩ Sp+1−q +⇒q+r Aq ∩ Sp+1−q = Or, +i.e. all q+r-facets through Aq are cut by Sp+1−q in r-flats through O. These flats +then cut the hypersphere in r-1-dimensional regions (the q+1-facets through Aq +cut the hypersphere in points, the q+2-facets cut it into great circles, etc.) +11 + +S3 +A +A=0Thus, the figure formed on the hypersphere is a regular tessellation on elliptic +space of p − q dimensions. If we denote by aNbc the number of b-facets passing +through a c-facet and lying on an a-facet of the general polytope and by aN ′ +bc +the configurational numbers of the regular tessellation of p − q dimensions, we +get +aN ′ +bc = a+qNb+q,c+q +When q = p − 1, a 1-dimensional elliptic regular tessellation can be obtained, +i.e. a regular polygon, and p+1Np,p−1 = 2N ′ +10 = 2. When q = p − 2, a 2- +dimensional elliptic regular tessellation is obtained as p+1Np,p−2 = 3N ′ +20 = +3N ′ +10 = p+1Np−1,p−2. The numbers +p+1Np,p−2 = p+1Np−1,p−2 = kp (p = 1, 2, ..., n − 1) +are analogous to the ones in the 4-dimensional case. For p = 1, the convention +rNp,−1 = Npr is used to denote the total number of p-facets in an r-facet. +Furthermore, denote the configurational numbers of the n-1-facet of the tessel- +lation by pFqr and the configurational numbers of the vertex figure by pVqr. +p+1Fp,p−2 = p+1Np,p−2 = kp ⇒ k1, k2, ..., kn−2 +p+1Vp,p−2 = p+2Np+1,p−1 = kp+1 ⇒ k2, k3, ..., kn−1 +Now, the geometrical method for distinguishing among the elliptic, Euclidean, +and hyperbolic case has to be generalized for any dimension. Consider a regular +tessellation k1k2k3...kn of n-dimensional space. Let C0 be any vertex, C1 the +midpoint of an edge though C0, C2 the center of a face (polygon) through C0C2 +and in general Cr the center of the r-facet through C0C1C2...Cr−1. Then the +triangle CpCqCr (p < q < r) is always a right angled triangle with right angle at +Cq. The angle C0C2C1 is half the angle at the center of a plane face subtended +by an edge and is equal to π/k1. The angle C1C0C2 = θ1 is half the angle +between two adjacent edges, C2C1C3 = θ2 is half the dihedral angle between +two adjacent plane faces and in general CpCp−1Cp+1 = θp is half the dihedral +angle between two adjacent p-facets. Since there are kn n-1-facets at each n- +2-facet, the angle Cn−1Cn−2Cn = θn−1 = π/kn. +Now consider a sphere at +C0 in the 3-facet C0C1C2C3 which is cut by the lines and planes through C0 +in a regular spherical polygon with sides 2θ1 and angles 2θ2 and the polygon +subtended at the center of the polygon by half the side is π/k2. +Hence by +spherical trigonometry +cosπ/k2 = sinθ2cosθ1. +Again, in the 4-face C0C2...C4 take a hyperplane H perpendicular to C0C1 at +C1 and in this hyperplane consider a small sphere centered at C1. It is cut +by the lines and planes in which H cuts the planes and hyperplanes through +C0C1 in a regular spherical polygon with sides 2θ2 and angles 2θ3 and the angle +12 + +subtended by the center of the polygon by half the side that is π/k3. By spherical +trigonometry again +cosπ/k3 = sinθ3cosθ2. +Finally we obtain the formula +cosπ/kr = sinθrcosθr−1 (r = 2, 3, ..., n − 1) with θn−1 = π/kn. +To determine θ1, consider the right triangle C0C1C2. In Euclidean geometry +the sum of its angles θ1 + π/k1 + π/2 = π and θ1 = π/2 − π/k1 while in elliptic +geometry it is more and in hyperbolic less. +Now for n = 5 take regular tessellation in 4-dimensional space. Then +cosπ/k3 = sinθ3cosθ2 = sinπ/k4cosθ2 and cosπ/k2 = sinθ2cosθ1 +sin2θ2 + cos2θ2 = 1 ⇒ cos2θ1sin2θ2 +cos2θ1 ++ sin2π/k4cos2θ2 +sin2π/k4 += 1 +cos2π/k2 +cos2θ1 ++ cos2π/k3 +sin2π/k4 += 1. +If the last expression is greater than 1, the tessellation will be elliptic, and for +less than one it will be hyperbolic. +Using these generalization further for n = 5, we require that the 4-facets and ver- +tex figures of a regular 5-polytope must be elliptic tessellations of 4-dimensional +space. Since in four dimensions the only such are the six regular polychora 333, +334, 335, 343, 433, 533 the only possible 5-dimensional cases are +3333 +3334 +3335 +3343 +3433 +4333 +4334 +4335 +5333 +5334 +5335 +Applying the condition +cos2π/k2 +cos2θ1 ++ cos2π/k3 +sin2π/k4 += 1 +we can divide the above cases into three elliptic 3333, 3334, and 4333, three +Euclidean 3343, 3433, and 4334 and the other five hyperbolic. Therefore, we +can conclude that there are three regular tessellations in an Euclidean space +of 4-dimensions, i.e. such space can be completely filled with 8-cells (cubes), +16-cells, or 24-cells and that there are only three regular 5-polytopes - the 5- +simplex, the 5-cube, and the 5-orthoplex. +For n = 6 since the 5-facet k1k2k3k4 and vertex figure k2k3k4k5 must both be +elliptic, the tessellations can only be of the form +13 + +33333, 33334, 43333, 43334 +which correspond respectively to 6-simplex, 6-cube, 6-othoplex and finally a reg- +ular tessellation of a 5-dimensional Euclidean space. It can immediately be seen +that in any further dimension the choice will remain the same and the self-dual +n-simplex αn, the n-cube γn and its dual n-orthoplex βn are the only regu- +lar polytopes in n dimensions (n ≥ 5) while the 43n−24 tessellation δn+1 is the +only regular tessellation of an n-dimensional space (notation due to Coxeter [3]). +3 +The quasi-regular and uniform tessellations +and polytopes +3.1 +Quasi-regular tilings and the crystallographic restric- +tion +The numbers obtained for each specific regular n-polytope in the previous sec- +tion turn out to be very essential for the description of its properties. They +will reappear in their specific combination later in this text as the orders of the +generators for Coxeter systems and in the classification of the edges of Coxeter +graphs. Here we first introduce the most basic notation which was briefly used +above called the Schl¨afli symbol and its extension for quasi-regular tessellations +first suggested by Coxeter [3]. For the sake of brevity this extension is analyzed +mainly for the case of polyhedra and respectively tilings and a final extension to +uniform tessellations then is introduced with an emphasis on honeycombs and +polychora. Nevertheless, quasi-regular honeycombs are briefly introduced in the +next section for completeness. Finally, since this subsection will emphasize on +tilings, its second part analyses the possible symmetries of all tilings. +The Schl¨afli symbol of a polytope with the numbers k1k2...kn−1 simply has the +form {k1, k2, ..., kn−1} and can therefore be used for all derived regular poly- +topes or tessellations. +Therefore, the Schl¨afli symbol of the n-1-facet of an +n-polytope {k1, k2, ..., kn−1} is {k1, k2, ..., kn−2}, the Schl¨afli symbol of the ver- +tex figure is {k2, ..., kn−1}, and the Schl¨afli symbol of the dual of the polytope +is {kn−1, kn−2, ..., k1}. +This notation can now be extended to quasi-regular polyhedra. The interior of +the intersection of two dual regular polyhedra {p,q} and {q,p} centered at the +same point has N1 vertices which are exactly the mid-edge points of both {p,q} +and {q,p}. Its faces consist of both N0 {q} and {p} polygons which are the +vertex figures respectively of {p,q} and {q,p}. There are 4 edges at each vertex +and 2N1 edges altogether. Then +N0 − N1 + N2 = N1 − 2N1 + (N0 + N2) = 2 +14 + +and the resulting polyhedron can be denoted as +� +p +q +� += +�q +p +� +. +The possible cases can be derived based on the above restriction: +p = q = 3 ⇒ N ′ +0 = N1 = 6, N ′ +1 = 2N1 = 12, N ′ +2 = N0 + N2 = 4 + 4 = 8 +p = 3, q = 4 ⇒ N ′ +0 = N1 = 12, N ′ +1 = 2N1 = 24, N ′ +2 = N0 + N2 = 6 + 8 = 14 +p = 3, q = 5 ⇒ N ′ +0 = N1 = 30, N ′ +1 = 2N1 = 60, N ′ +2 = N0 + N2 = 12 + 20 = 32 +For the three elliptic cases we then get +�3 +3 +� += {3, 4}−octahedron, +�3 +4 +� +−cuboctahedron, and +�3 +5 +� +−icosidodecahedron +and since the edges are all alike, each separating p from q, this gives rise to +the definition of a quasi-regular polyhedron as a polyhedron with exactly two +kinds of regular faces that is edge-transitive. The proof that these are the only +quasi-regular polyhedra originates from the fact that the dihedral angles at a +vertex make a total that must conform to the inequality +r(1 − 2 +p)π + r(1 − 2 +q )π < 2π. +Therefore, +1 − 2 +p + 1 − 2 +q < 2 +r ⇒ 1 − 1 +p − 1 +q < 1 +r ⇒ 1 +p + 1 +q + 1 +r > 1 +and since p and q cannot be less than 3, r = 2 and p = 3, q = 4 or p = 3, q = 5. +In addition we can consider the dual pair of regular tilings {3,6} and {6,3} where +the trihexagonal tiling +�3 +6 +� +is obtained with vertices as the intersections of the +their edges. +The symmetry group of such a tiling is an infinite group of congruent trans- +formations in the plane. This group contains a finite subgroup of index 2. A +classification of the symmetries of the plane tilings then leads to the crystallo- +graphic restriction theorem which states that if a discrete group of rotations in +the plane has more than one center of rotation, then the only rotations that can +occur are of order 1, 2, 3, 4, and 6. +Consider the Euclidean motion group R2.O(2) of isometries on the plane. Any +finite subgroup of this group fixes a point and so is conjugate to a finite subgroup +of O(2) that fixes the origin. The finite subgroups of O(2) are then the cyclic +groups of order n, i.e. rotations of 2π/n, and the dihedral groups of order 2n +with rotations as a subgroup of index 2 and reflections that conjugate those to +their inverse rotations. The subgroup R2 consists of all translations. Consider +all discrete cases, i.e. rotations and translations that cannot be arbitrarily close +to the identity transformation and are bounded from below. Let Γ ≤ O(2) be +15 + +discrete subgroup and consider the lattice group L = Γ∩R2 = Za+Zb generated +from translations by two linearly independent vectors. Furthermore, let ¯Γ be +the image of R2.O(2) in R2.O(2)/R2, i.e. ¯Γ = Γ/L. +Then ¯Γ preserves the lattice L. The proof is the following. Take a vector b ∈ L +or equivalently a translation by this vector tb ∈ Γ and take γ ∈ Γ that maps to +¯γ ∈ ¯Γ. In terms of linear operators set +γ(v) = Av, tb(v) = v + b, γ−1(v) = A−1v +γtbγ−1(v) = γtb(A−1v) = γ(A−1v + b) = A(A−1v + b) = v + A(b) +Therefore γtbγ−1 = t¯γ(b) is a conjugate translation by ¯γ(b) and since γ, tb ∈ Γ +by the operation closure it follows that t¯γ(b) ∈ Γ and finally ¯γ(b) ∈ L. +Because of this established fact, ¯Γ = Cn or ¯Γ = D2n with n = 1, 2, 3, 4, 6 and +¯Γ = Cn as rotation parts has maximum order of 12. A proof in terms of linear +operators is as follows. Let A ∈ ¯Γ be a rotation, i.e. detA = 1. We have to +show that the order of A is 1,2,3,4 or 6. Consider the characteristic polynomial +of A x2 − tr(A)x + det(A) = x2 − tx + 1. Consider A to be a rotation by θ. +Then since rotation is analogical to scaling with complex numbers, x2 − tx + 1 +has complex roots, i.e. t2 − 4 ≤ 0 (the only real cases are +1 and −1). Thus, +the matrix is diagonalizable over the complex numbers and tr(A) = t = z + ¯z is +a real number. Furthermore, since A stabilizes the lattice L = Za + Zb, it takes +both a and b to integer multiples of a, b. If we take these as column vectors of +A in the basis a, b, A will have integer entries in this basis and thus its trace +t is an integer. Then t2 − 4 ≤ 0 ⇒ t = ±1, ±2, 0 where t = ±2 = 2cos(θ) are +the rotations of order 1 and 2, t = ±1 = 2cos(θ) are the rotations of order 3 +and 6, and t = 0 = 2cos(θ) are the rotations of order 4. This concludes the proof. +Considering Gaussian integers for coordinates, the symmetry group of the tiling +{4,4} is then generated by the translation z′ = z + 1 and the rotation z′ = iz +of order 4 while the symmetry group of the {3,6} is generated by the same +translation along with a rotation z′ = eπi/3z = (e2πi/3 + 1)z of order 3. The +crystallographic restriction theorem can be used to classify all symmetries of a +tiling on the plane with translations, rotations, reflections, and glide reflections +as isometries of the Euclidean plane, also called wallpaper groups or crystallo- +graphic groups on the plane. +3.2 +Uniform honeycombs and geometric operations +The extension of the Schl¨afli symbol introduced by Coxeter can be generalized +for rectified n-polytopes where rectification is the process of taking the inter- +section of two dual polytopes, i.e. cutting the vertices at the midpoints of the +edges which will result in a polytope bounded by both the vertex figures and +rectified faces of the original. +16 + +For a quasi-regular honeycomb, all cells must be regular and all vertex figures +must be quasi-regular. Alternative conditions then are that the vertex figures +are all the same and the cells are of two alternating kinds. The only two regular +polyhedra, whose angle sum divides 2π are the octahedron and the tetrahedron +(their sum is π). The only quasi-regular honeycomb then is +�3, 3 +4 +� +or the al- +ternated cubic honeycomb. It can be seen as a cubic honeycomb with alternate +vertices removed reducing cubic cells to tetrahedra and creating octahedron cells +in the remaining gaps. +Rectification is a special case of the more general truncation operation which +can be used to derive a list of uniform polytopes and tessellations. Although +there is a new notation known as the Wythoff symbol originating from the +Wythoffian construction of uniform polytopes, there is also a further extension +of the Schl¨afli symbol and a notation that will be introduced in the next section +that adds more information about the symmetries of the tessellation (resp. of +the polytope). The final extended Schl¨afli symbol denotes the k-th rectifica- +tion of a polytope as tk{p1, p2, ..., pn−1} where t0,1 is a truncation applied to +polygons or polytopes of higher dimension, t0,2 is cantellation (both edges and +vertices removed) applied to polyhedrons or higher, t0,3 is runcination, t0,1,2 is +cantitruncation (cantellation and trucation), t0,1,2,3 is runcicantitruncation, etc. +All these geometric operations can be generalized in terms of sequences of cross- +sections with higher dimensional space. Truncation results from the sequence +of cross-sections parallel to a facet of the vertex figure of the polytope where +crossing the vertex leads to intersection of the previously crossed edges and +origination of the next. Cantellation is analogously the result of sequence of +cross-sections parallel to an edge of the polytope where crossing the edge leads +to intersection of previously crossed faces and origination of the next. Such a +generalization can be used for the reconstruction of an n + 1-dimensional poly- +tope through the geometrical operations performed on its cross-sections. +Using this, we were able to reconstruct the graph of a 4-polytope with cross- +sections which are the gradual truncations of a cube to its dual. +These operations (besides rectification) introduce the semi-regular polytopes in +addition to the quasi-regular and regular ones. All of them are unified under +the definition of a uniform polytope whose only two conditions are: (1) uniform +polytope facets; (2) vertices of the same kind (vertex-transitivity). The uniform +2-polytopes are necessarily the regular polygons. +There are 11 uniform tilings of the plane and 28 uniform convex honeycombs in +3 dimensions, also called the Archimedean honeycombs. From the latter, there +are just one regular (cubes) and one quasi-regular (octahedra and tetrahedra), +both mentioned above. Truncation has been used to derive 7 additional hon- +eycombs originating from the cubic one and 4 additional originating from the +17 + +alternated cubic honeycomb. Finally there are 15 more from prismatic forms +derived from modifications of the uniform plane tilings. +4 +Coxeter groups and reflection groups +4.1 +Coxeter systems +In the previous section it was shown that the group of isometries on the Eu- +clidean plane admits only a few discrete subgroups of O(2) thus allowing a +classification of the possible symmetries of a tiling that is not necessarily uni- +form. The 3-dimensional case of crystallographic groups is the space groups. +A formal description of the symmetries of a tessellation (resp. polytope) is the +goal of this section which leaves the techniques of n-dimensional geometry aside +and uses tools mainly from abstract algebra. +A Coxeter system is a pair (W, S) where W is a group and S a set of generators +S ⊂ W restricted by the relations (sisj)m(si,sj) = 1 where m(si, si) = 1, i.e. si +is an involution, and m(si, sj) = m(sj, si) ≥ 2, ∀si ̸= sj ∈ S. If there is no +relation between si and sj the convention m(si, sj) = ∞ is used. The group W +is then the quotient F/N where F is a free group on the set S and N is the +normal subgroup generated by all elements (sisj)m(si,sj). Furthermore, |S| = n +is the rank of the Coxeter system. W(M) is then a Coxeter group that can be +constructed from a symmetric n × n matrix M = (mij)1≤i,j≤n indexed by S +with entries in Z ∪ {∞} such that mii = 1 and mij ≥ 2 ∀i ̸= j. The Coxeter +group of type M is then analogously +W(M) = ⟨s1, s2, ... ∈ S|(sisj)mij = 1, mij ∈ M⟩ +which will be denoted W when no ambiguity is possible. +A few lowest rank examples are the following. If |S| = n = 1 then M = (1) +and W(M) = ⟨s1|s2 +1 = 1⟩ which is the cyclic group of order 2. +For n = 2 +then M = +� 1 m +m 1 +� +⇒ W(M) = ⟨s1, s2|s2 +1 = 1, s2 +2 = 1, (s1s2)m = 1⟩ = D2m for +m ∈ N∪∞ which is the Klein Four group for m = 2, the dihedral group of finite +order for 2 ≤ m ≤ ∞ and the infinite dihedral group D∞ for m = ∞. +Instead of the matrix, the Coxeter system (W,S) can be constructed from an +undirected graph Γ with a vertex set S where two vertices s and s′ are joined +with an edge that is labeled m(s, s′) if 3 ≤ m ≤ ∞. Therefore if the distinct ver- +tices s and s′ are not joined, then m(s, s′) = 2. The edges with label m(s, s′) = 3 +are omitted due to their frequency and by convention. The resulting graph Γ +is called Coxeter graph and as a notation contains more information than the +extended Schl¨afli symbol. +18 + +Figure 4: Sample Coxeter graph. +The entire information about the matrix can be reconstructed from the Coxeter +graph as the same applies to the Schl¨afli symbol. To show this we need to study +the connection of the Coxeter groups to the reflection groups and the groups of +symmetries of the regular polytopes. +Because Coxeter groups are based on an abstract presentation, they do not +necessarily admit a faithful representation as reflection groups. The abstract +group of each reflection group is a Coxeter group as the reflections are a special +case of involutions [3]. Infinite Coxeter groups in particular may not admit a +representation as a reflection group. However, finite Coxeter groups have a faith- +ful linear representation as groups generated by reflections in Euclidean space. +Some of the finite linear groups generated by such reflections are groups of sym- +metries of the regular polytopes in Euclidean space. We will show that Coxeter +groups admit a representation since this is essential for the purpose of this work. +Let (W, S) be a Coxeter system of type M and take n = |S|. We will now +construct a real linear representation of W of degree n such that the images of +the elements of S are reflections in Rn. A reflection on a real vector space V is a +linear transformation on V fixing a subspace of V of codimension 1, a reflection +hyperplane Hα, and having a nontrivial eigenvector α with eigenvalue λ = −1, +called a root of the reflection. Now take the vector space V over the field R +with a basis {αs|s ∈ S}. Define a bilinear form B on V such that +B(αs, αs′) = −cos +π +m(s, s′) +where m(s, s′) = mij ∈ M and since mij = mji the form is symmetric. Then +B = −1 for m(s, s′) = ∞, B(αs.αs) = 1, and B(αs.αs) ≤ 0 for s ̸= s′ +with equality only for m(s, s′) = 2. +Now for each s ∈ S define a reflection +σs : V → V such that σsλ = λ − 2B(αs, λ)αs. Then σsαs = −αs and σsλ = λ +with {λ ∈ V |B(x, λ) = 0} := Hs the hyperplane orthogonal to αs. Therefore +σs has order 2 in GL(V ). Additional observation from the definition is that +B(σsλ, σsµ) = B(λ, µ) ∀λ, µ ∈ V , i.e. the reflection preserves the bilinear form +and each element generated by σs will preserve it. A final observation is that +|σsσs′| = m(s, s′) and should be proven next. +Consider Vs,s′ := Rαs⊕Rαs′. The restriction of B to Vs,s′ is positive semidefinite +19 + +. +1 +2 +n-1 +nsince for λ = aαs + bαs′ we get +B(λ, λ) = B(aαs + bαs′, aαs + bαs′) = += a2B(αs, αs) + 2abB(αs, αs′) + b2B(αs′, αs′) = += a2 − 2abcos(π/m(s, s′)) + b2 = += a2 − 2abcos(π/m(s, s′)) + b2(cos2(π/m(s, s′)) + sin2(π/m(s, s′)) = += (a − bcos(π/m(s, s′)))2 + b2sin2(π/m(s, s′)) ≥ 0. +Therefore the form is positive definite if sin(π/m(s, s′)) ̸= 0 and m < ∞. Note +further that σs and σs′ stabilize Vs,s′, so the order of σsσs′ as an operator in Vs,s′ +has two cases. (1) If m < ∞ since the form is positive definite we can consider +Euclidean plane and since B(αs, αs′) = −cos(π/m(s, s′)) = cos(π−(π/m(s, s′))) +and the angle between Rαs and Rαs′ is therefore π −(π/m(s, s′)), the angle be- +tween Hα and Hα′ is π/m(s, s′). Because rotation through 2π/m(s, s′) can be +achieved as a product between two reflections with an angle between their fixed +hyperplanes π/m(s, s′), it follows that σsσs′ has order m(s, s′). The fact that B +is positive definite on Vs,s′ implies that V = Vs,s′ ⊕ V ⊥ +s,s′ where V ⊥ +s,s′ := s⊥ ∩ s′⊥ +is fixed by both σs and σs′. Then σsσs′ has order m(s, s′) also on V . (2) If +m = ∞, B(αs, αs′) = −1. Then if λ = αs + αs′, B(λ, αs) = B(λ, αs′) = 0 +and σs and σs′ fix λ. Then σsσs′αs = σs(αs + 2αs′) = 3αs + 2αs′ = 2λ + αs. +Applying σsσs′ k ∈ Z times, (σsσs′)kαs = 2kλ + αs. Therefore we can conclude +that σsσs′ has infinite order on Vs,s′ and therefore also on V. +This helps us conclude that there is a unique homomorphism σ : W → GL(V ) +where σ(s1s2...sn) = σ(w) = σw = σs1σs2...σsn. We call this homomorphism a +linear representation of W. The fact that this representation is then faithful is +a corollary from another theorem regarding the length function for the Coxeter +group and is outside of the scope of this text. However, one final definition which +is necessary for the later sections is as follows. The subgroup WI generated by +a given subset I ⊂ S and any of its conjugates are called parabolic subgroups +of the Coxeter group W and are in fact Coxeter groups themselves. +4.2 +Root systems +Consider a Coxeter system (W, S) of type M. The set Φ := � +s∈S,w∈W σ(w)αs +is called the root system of W. It consists of the collection of orbits of the +unit vectors in the reflection representation space V on which W acts, i.e. the +collection of unit vectors in V permuted by W. +These remain unit vectors +because W preserves the bilinear form. Furthermore, Φ = Φ+ ∪ Φ− where +Φ+ := Φ ∩ ( +� +s∈S +R≥0αs), Φ− := Φ ∩ ( +� +s∈S +R≤0αs). +A root α is called positive (write α > 0) if α ∈ Φ+ and negative (α < 0) if +α ∈ Φ−. Note that Φ = −Φ since σs(αs) = −αs, Φ ∩ Rα = {α, −α} ∀α ∈ Φ, +and σ(W)Φ = Φ. Positive and negative roots are defined with regard to certain +20 + +total ordering like lexicographic ordering where � aiλi < � biλi ⇒ ak < bk +where k is the least index such that ak ̸= bk. +A subset ∆ ⊂ Φ+ of vectors αi constituting a basis for the R-span of Φ in V is +called a simple system. A root α ∈ ∆ is called simple root of Φ. Each α′ ∈ Φ is +a linear combination of ∆ with coefficients all of the same sign. +Further relation between the roots and the reflections can be established through +the consideration of the set R = {wsw−1|w ∈ W, s ∈ S} or reflections of the +Coxeter system (W, S). By the geometric representation σ : W → GL(V ), each +s ∈ S acts on V as a reflection σ(s)αs. More generally, a reflection in GL(V ) +can be associated to each root α ∈ Φ. Consider α = σ(w)αs := w(αs) (for +brevity) for w ∈ W and s ∈ S. Then wsw−1 ∈ R acts on W as follows: +σ(wsw−1)λ =: wsw−1(λ) = w[w−1(λ) − 2B(w−1(λ), αs)αs] = += λ − 2B(w−1(λ), αs)w(αs) = λ − 2B(λ, w(αs))w(αs) = += λ − 2B(λ, α)α +which shows that wsw−1 does not depend on the choice of w and s but only on +the choice of α so can be denoted sα. Furthermore, sα acts on V as a reflection +sending α to −α, fixing pointwise the hyperplane Hα ⊥ α. As a consequence, α +and −α both determine the same reflection sα = −sα. The root-reflection corre- +spondence is finally established due to the bijective map α → sα (for α ∈ Φ+). +Therefore, each reflection of a Coxeter group W has a unique positive root +α ∈ Φ+ and each α ∈ Φ+ is the positive root of a unique orthogonal reflection +with respect to B. If α, β ∈ Φ and w, w′ ∈ W such that w′(β) = w(α), then +wsαw−1 = w′sβw′−1. +This correspondence helps us interpret the relations in a Coxeter system of the +form (sisj)mij as the result of two reflections fixing hyperplanes meeting at an +angle π/mij. The element sisj ∈ S being of order mij than has the geometrical +interpretation of a rotation by 2π/mij. +Some further important properties and definitions of root systems to be men- +tioned here are as follows. If ∆ is a simple system in Φ, then (α, β) ≤ 0 ∀α ̸= β ∈ +∆. If αs ∈ ∆ then s(Π\{αs}) = Π\{αs}. A root system Φ is crystallographic if +it satisfies the additional requirement +2(α, β) +(β, β) ∈ Z ∀α, β ∈ Φ. +The vector α∨ := 2α/(α, α) is called a coroot of α ∈ Φ. The set Φ∨ of all +coroots is the dual root system or the inverse root system of Φ. The reflections +w ∈ W generated by Φ∨ are the same as Φ, i.e. wα∨ = w(α)∨. +A final definition for this section should be given for the fundamental domain of +the action of W on V . Take a positive system Φ+ containing a simple system ∆ +21 + +and consider the open half-spaces H+ +α := {λ ∈ V |(λ, α) > 0} and H− +α := −H+ +α +of each reflecting hyperplane Hα. Define C := ∩αs∈∆H+ +α which is open and +convex as intersection of open and convex sets. Let D := ¯C be the closure of +C, i.e. the intersection of the closed half-spaces H+ +α ∪ Hα. Then +D = {λ ∈ V |(λ, α) ≥ 0 ∀α ∈ ∆} +and each λ ∈ V is conjugate to exactly one point in D. Thus D is called a +fundamental domain for the action of W in V . +This could offer a different insight into the geometric representation of the Cox- +eter group. The nodes of the Coxeter graph represent the walls of the fundamen- +tal domain and two nodes are joined by a branch whenever the corresponding +walls are not perpendicular. Moreover, the branches are marked with numbers +mij > 2 to indicate the angles π/mij. In the case of a connected graph with- +out any even marked branches, all the reflections in the group are conjugate +to one another [3]. This interpretation of the Coxeter graphs in terms of fun- +damental domains originally proposed by Coxeter leads to their final extension +to the geometrical operations discussed in the previous section and the uniform +tessellations and polytopes. +4.3 +Classification +All symmetry groups of regular polytopes are finite Coxeter groups (and resp. +finite reflection groups). All symmetry groups of regular tessellations are affine +Coxeter groups (and resp. affine reflection groups containing normal abelian +subgroups such that the quotient group is finite and is itself a Coxeter group). +The Coxeter graph of an affine Coxeter group is obtained by adding an addi- +tional vertex as in the construction performed in section 2. The dual polytopes +or tessellations have the same symmetry groups and therefore the same Coxeter +groups. +For the regular polytopes in any number of dimensions, the symmetry groups +are respectively the symmetric group Sn+1 or the Coxeter group of type An for +the regular n-simplex αn, and the hyperoctahedral group or the Coxeter group +of type Bn = Cn for the n-cube γn and its dual the n-orthoplex βn. The root +systems for these use the same notation with the difference that Bn and Cn are +interchanged (dual root systems with B2 and C2 isomorphic). The root system +Bn of γn which is specifically important for one of the algorithms described in +the next section is constructed as follows. +Let V = Rn, and let Φ consist of all integer vectors in V of length 1 (short +roots) or +√ +2 (long roots). The total number of roots is 2n2 with 2n short roots +±ei and 2n(n − 1) long roots ±ei ± ej(i < j). For ∆ take the simple long roots +αi = ei − ei+1, for 1 ≤ i ≤ n − 1, and the short root αn = en. The reflection s +through the hyperplane perpendicular to the short root αn is then the negation +22 + +of the n-th coordinate. +Furthermore, the symmetry group of the pentagon is H2, the symmetry group +of the dodecahedron and its dual icosahedron is the full icosahedral group H3 +and the symmetry group of their 4-dimensional analogues (the 120-cell and +the 600-cell) is H4. The symmetry group of the 24-cell is F4. The Coxeter +groups of type Dn (n-demihypercube), E6 (221, 122), E7 (321, 231, 132), and E8 +(421, 241, 142) are the symmetry groups of certain semiregular polytopes. The +symmetry group of the hexagon is G2. +The affine Coxeter groups are then classified as ˜ +An for the simplectic uniform +tessellation, ˜ +Bn for the demihypercubic uniform tessellation, ˜ +Cn for the hyper- +cubic uniform tessellation, ˜ +E6 for 222, ˜ +E7 for 331, 133, ˜ +E8 for 521, 251, 152, ˜F4 +for 16-cell and 24-cell uniform tessellations and ˜ +G2 for hexagonal and triangular +tiling. +The final notation to be reviewed in this work is the ringed Coxeter graph +which contains enough information to explicitly enumerate almost all types of +uniform polytopes and uniform tessellations. +Each uniform polytope can be +generated using the mirror hyperplanes bounding the fundamental region and +a single generator point. The reflections of the point through the mirror hy- +perplanes and their further reflections through the same hyperplanes form the +set of vertices of the polytope. The edges of the polytope connect each point +to a mirror point; the faces can be constructed as cycles of edges, etc. The +location of the generating vertex is specified as all nodes of the Coxeter graph +corresponding to the mirror hyperplanes on which the vertex does not lie are +marked with a ring (equidistant from all ringed node hyperplanes). Thus, all +mirror hyperplanes where the generating vertex lies do not generate additional +vertices. A diagram needs at least one active node to represent a polytope and +therefore all Coxeter graphs of the regular polytopes have their first node ringed. +The more general case of uniform polytopes with one marked hyperplane cor- +responds to a generating point at a vertex of the fundamental domain (which is +always a simplex because of the way it is constructed). If all nodes are ringed, +the generator point lies in the interior of the simplex. Generally, if n nodes of +the Coxeter graph are marked, the generating point gets n−1 degrees of freedom +on n-1-facets of the fundamental domain and can generally be taken to be at the +center the n-1-facet for equal n-1-faces of the final polytope. A secondary fea- +ture can be used for the special cases of uniform polytopes with non-reflectional +symmetry where the central dot of a ringed node is removed to imply alternate +nodes deleted. The constructed polytope will then only have a subsymmetry of +the original Coxeter group. Eventually if all nodes are marked in this way, the +polytope is called a snub. Using this final notation we can describe for exam- +ple the cuboctahedron, rhombicuboctahedron, octahedron, truncated cube and +other uniform polytopes derived from geometric operations on the cube. +23 + +5 +The hypercube tessellation and convex poly- +tope algorithms +5.1 +The developed algorithms +Two of the algorithms developed for this work will be presented in this section. +Since they consist of some common approaches and differ by removable compo- +nents, they will be described in terms of one sequentially consistent algorithm. +Although our initial plan was to use parametric equations, it offered no good +solution when it comes to the calculation of the intersection. Furthermore, al- +gebraic equations did not allow for plotting of vertical lines. +The hypercube tessellation algorithm generates an n-cube and n additional n- +cubes on all of its sides and obtains a cross-section with a 3-dimensional space +parallel to three independent vectors (preferably roots from the Bn root sys- +tem). This results in the 3-dimensional plot of a few space-filling polyhedra +that build up the tessellation and an additional optional 2-dimensional plot. +The convex polytope algorithm generates a convex n-polytope by analyzing its +possible k-facets from its vertex coordinates and then performs a similar cross- +section through specified parameters. +For an n-dimensional convex polytope with V as the set of its vertices and F +as the set of its n-facets, take all subsets S ⊂ V, |S| = n. For these n points +A1, A2, ..., An find a nonzero vector (u1, u2, ..., un) such that it is orthogonal to +the vectors Ai − An, i = 1, 2, ..., n − 1. Thus, we need to solve a homogeneous +system of n − 1 equations in n variables u1, u2, ..., un. The nullspace of the con- +structed matrix can be more than one dimension in case the n vertices constitute +a subspace of the hyperplane (there at least two linearly independent non-zero +vectors in the kernel of the considered linear map). These cases immediately +imply that the given S is not an n-1-facet of the polytope and therefore can be +ignored. +The remaining results are the normal vectors ⃗nj of the hyperplanes Hj passing +through the points from S where j indexes each of the remaining cases for S. It +is important to consider both positive and negative orientation of the ⃗nj when +Hj is tested for an n-face. For this purpose one of the definitions of convexity +for the polytope has been used. +A polytope is convex if it lies entirely on one side of each of its n-1-facets. +Therefore, if the polytope is convex and has a facet lying in the hyperplane Hj, +all points A ∈ V \S must lie in the closed negative half-space H− +j . The signed +distance is obtained from the projection of any vector from the hyperplane to the +24 + +point A = (a1, a2, ..., an) onto the normal vector of the plane ⃗nj = (u1, u2, ...un). +|(⃗a − ⃗x) · ⃗nj| +| ⃗nj| += |a1u1 + a2u2 + ... + anun − u1x1 − u2x2 − ... − unxn| +| ⃗nj| += += |a1u1 + a2u2 + ... + anun − (−C)| +| ⃗nj| += a1u1 + a2u2 + ... + anun + C +� +u2 +1 + u2 +2 + ... + u2n +The translation constant C can be obtained from the regular hyperplane equa- +tion of Hj evaluated at any point A ∈ S. The convex hull defined by the vertices +in V is then +� +H− +jk = conv(V ), k = 0, 1, ..., |F| +which is the desired polytope and the solution of |F| inequalities. +However, only the normal vectors njk of the n-facets of the polytope are nec- +essary for the second part of the algorithm which is the cross-section with a +3-subspace where we can observe certain symmetries of the polytope. Three +linearly independent vectors and a translation point are sufficient for defining +a unique 3-subspace inside the n-space. In order to observe symmetries of the +polytope in the 3-subspace, only specific orientations are allowed. These are +determined by the root system Φ of the Coxeter group for the specific polytope. +Thus, picking a 3-subspace which is parallel to three linearly independent roots +(not necessarily simple) ensures that the observed cross-section conforms to the +symmetries of a parabolic subgroup of the original Coxeter group. It then can +in term be described with the root system stabilized by this subgroup. +For +example considering the root system B4 and a hypercube, we can pick the long +roots +u1 = (-1,-1,0,0) +∠(u1, u2) = π/3 +u2 = (-1,0,-1,0) +∠(u2, u3) = π/4 +u3 = (-1,0,0,0) +∠(u3, u1) = π/4 +which will result in B3 root system and the cross-section with the hypercube +should be invariant under all reflections along these roots. The example cross- +section is a cube standing on its vertex with respect to the xy-plane and the +resulting 2-dimensional cross-section is a hexagon. Therefore, instead of picking +any possible orientation of the 3-subspace which is also possible, we pick only +vectors α ∈ Φ. +It is important to note that the choice of a normal vector of the 3-subspace is +not unique if its codimension in the n-space is greater than 1. For a larger codi- +mension, i.e. larger dimension of the orthogonal complement of the 3-subspace, +we would have to select a basis of n−3 vectors in order to uniquely identify the +3-subspace. Therefore, a much better approach is to simply select three vec- +tors spanning the 3-subspace in order to determine its orientation in the n-space. +25 + +After the three roots are selected, the Gram-Schmidt orthogonalization process +can be used to produce the new orthonormal basis and a transition matrix T +with the unit vectors as columns vectors. Once the linearly independent roots +are chosen, orthogonalization inside the 3-subspace naturally does not influ- +ence the cross-section in any way. The n-facet normal vectors T −1njk = T tnjk +then define new positive half-spaces which reorients the convex polytope. In +the case of the hypercube tessellation algorithm, the same transformation is +applied to all side-cubes with the difference that they are initially translated +along some initial unit vector. When the half-space inequalities are produced, +translation of the hyperplanes is applied using the point specifying the trans- +lation of the 3-subspace. Finally, the inequalities are solved for x1, x2, and x3 +given xi = 0, 3 < i ≤ n and the result is ready for plotting. +5.2 +Obtained results +The following subsection introduces some important observations and results +obtained in this work from both the algorithms and the performed study. +Consider the vector subspaces K ⊂ L ⊂ M where dimK = k, dimL = l, and +dimM = m < n. Take a sequence of m linearly independent roots from Φ which +uniquely determine M, such that the first l of them uniquely determine L and +the first k of them uniquely determine K. The permutation of the m roots +leads to different orientation of the cross-section with the n-polytope in M and +to different selection or permutation of the k roots for K and of the l roots for +L. Consider the permutations of m roots that stabilize the k roots in K but do +not stabilize the l roots in L. As a result the cross-sections of the polytope in +L are changing but the cross-sections with K remain the same. This provides +with a good example of multiple polytopes in l dimensions that have the same +k cross-section. For this same reason, a cross-section of a plane through the +origin of a specifically oriented octahedron or cube both result in a hexagon. In +the same way the observed polytope in the 3-subspace remains unchanged while +many different orientations of the original polytope and cross-section polytopes +can be observed in higher dimensions. +Another result of even higher importance is related to hidden symmetries of +the n-polytope that could not be obtained through its root system. Choosing +three independent roots guarantees that the cross-section will pertain at least +the symmetries resulting from these roots i.e. will be invariant under reflection +through the hyperplanes orthogonal to the roots. However, the cross-section +might have additional symmetries and could remain invariant under additional +reflections that do not preserve the original n-polytope. We can easily illustrate +this with an example. +A cross-section of 3-cube with a 2-flat that intersects the middle points of six of +its edges results in a hexagon. The symmetry that should formally be observed +26 + +is the one of the D6 group since all roots of the cube that are parallel to the 2-flat +are long and generate A2 root system. The angle between two roots parallel to +the 2-flat is 2π/3 and thus reflection along their respective hyperplanes (planes +in this case) result in a 3-fold rotation that will preserve the cube. Another +cross-section that is parallel to these two roots and intersects three vertices +of the original cube gives an equilateral triangle where the extra symmetry is +already not present. In the case of the initial cross-section, the roots of the A2 +root system all point to the vertices of the hexagon. As an even-sided polygon, +the hexagon is also invariant under reflection defined by another root which +points to the middle of its side. +Figure 5: Hidden symmetry in the cube. +Denote the two simple roots of the G2 root system α (pointing to a vertex +and coinciding with a root from the required A2 root system) and β (point- +ing to the middle of a side). After observing the net of the cube, it can eas- +ily be concluded that all roots of the hexagon of type β point to coordinates +(k/3, l/3), k = 1, 2, l = 1, 2 on a given face of the cube, a total of four for each +side. A general reflection of the cube along the β vector fixes the orthogonal +plane Hβ crossing two opposite vertices and the middle points of two edges. It +does not preserve the cube and therefore does not belong to the symmetry group +of the cube. As a result the β vector cannot be a root of the B3 root system and +the D12 group as the symmetry group of the hexagon is not a subgroup of the +octahedral group Oh. Parabolic subgroups of the Coxeter group of a hypercube +thus do not encompass all symmetries that might be observed in a cross-section +with the hypercube and selecting roots from its root system only ensures the +minimum symmetry of the particular cross-section. +Symmetry of larger order can be obtained through cross-sections of higher di- +mensions in the same way. A central cross-section with a 4-cube that is parallel +to three independent long roots generating a root system A3 results in a specif- +ically oriented octahedron (instead of a tetrahedron) which remains invariant +under reflections along all roots in A3. Hidden symmetries are also the rea- +27 + +son why uniform polyhedra like the cuboctahedron cannot directly be obtained +from intersections with an n-cube. Cuboctahedron can easily be obtained from +a central cross-section with a 16-cell (4-orthoplex) that is parallel to three short +roots of the Cn root system and 16-cell can be obtained from a 4-dimensional +central cross-section with the 5-cube. However, any further permutation of the +sequence of four roots which are sufficient for a 16-cell cross-section will result +in a +�4 +3 +� +possible 3-dimensional cross-sections none of which will take advantage +of the additional symmetry of the 4-dimensional cross-section necessary for ob- +taining the cuboctahedron. +A final conclusion then concerns the 3-dimensional cross-sections that can be +obtained from an n-dimensional cubic tessellation. Rotating the tessellation will +result in different cross-sections with its space-filling hypercubes and therefore +different additional symmetry of the resulting space-filling polyhedra. Returning +to the simple 3-dimensional example, the squares tiling can be obtained from +a face-first intersection with the cubic honeycomb and the equilateral triangles +from a vertex-first cross-section through three of the vertices of any of the cubes. +The hexagon tiling however can be achieved only if the cubes are interpreted +as polyhedra with additional vertices at the center of each original face with +solid angles of π and the original cubes are then arranged in a way so that the +vertex of a cube touches the center of the original face of another and its edges +remain parallel to the edges of the other. If the original cubic honeycomb is +retained and at least one of the cubes is intersected at the centers of six of its +edges, the resulting tessellation will be the trihexagonal tiling of the plane which +consists of polygons with additional symmetry and polygons with the minimum +necessary symmetry. +28 + +Appendices +A +Visualizations +Figure 6: The convex polytope algorithm - rhombic dodecahedron obtained +from 24-cell cross-section and hexagonal bipyramid obtained from 16-cell cross- +section. +Figure 7: The hypercube tessellation algorithm - trihexagonal tiling and alter- +nated cubic honeycomb. +29 + +References +[1] B. Gruenbaum and G. C. Shephard, Convex Polytopes. Bulletin London +Mathematical Society, Oxford, 1 (1969), 257-300. +[2] D. M. Y. Sommerville, An Introduction to the Geometry of N Dimensions. +Dover Publications, New York, 1st Edition, 1958. +[3] H. S. M. Coxeter, Regular Polytopes. The Macmillan Company, New York, +2nd Edition, 1963. +[4] James E. Humphreys, Reflection Groups and Coxeter Groups. Cambridge +University Press, Cambridge, 1st Edition, 1990. +30 + diff --git a/t9AzT4oBgHgl3EQf6v6W/content/tmp_files/load_file.txt b/t9AzT4oBgHgl3EQf6v6W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e8b22b28d2501d4c53da16f7459f97ef7ba616f --- /dev/null +++ b/t9AzT4oBgHgl3EQf6v6W/content/tmp_files/load_file.txt @@ -0,0 +1,849 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf,len=848 +page_content='American University in Bulgaria Senior Thesis Tessellations Author: Plamen Dimitrov Advisor: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Orlin Stoytchev January 6, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='01880v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='GR] 5 Jan 2023 Abstract This work presents the tessellations and polytopes from the perspective of both n-dimensional geometry and abstract symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It starts with a brief introduction to the terminology and a short motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the first part, it engages in the construction of all regular tessellations and polytopes of n dimensions and extends this to the study of their quasi- regular and uniform generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the second part, the symmetries of polytopes and tessellations are considered and the Coxeter groups and their associated root systems are introduced and classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the last part, the algorithms developed for this work are described and their results discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 1 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 Motivation and tessellation definitions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 4 2 The regular tessellations and polytopes of n dimensions 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 The regular tilings and polyhedra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': 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Uniform honeycombs and geometric operations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 22 5 The hypercube tessellation and convex polytope algorithms 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 The developed algorithms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2 Obtained results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 26 Appendices 29 A Visualizations 29 Bibliography 30 2 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 Motivation and tessellation definitions The idea for this work started from the visualization of the sequence of 4-cube cross-sections with a 3-flat (3-dimensional affine space) which results in an oc- tahedron at the center of the 4-cube and in tetrahedrons of different size and truncation magnitude along any of the 4-fold diagonals of the 4-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Figure 1: 3-dimensional cross-sections along 4-fold symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The fact that the three regular polytopes in all dimensions could be obtained from a cross-section with a hypercube and that various uniform polytopes can further be obtained from these together with the fact that the hypercubic tes- sellation is the only regular tessellation in all dimensions led to the idea to develop an algorithm that could help us identify specific types of symmetric cross-sections with the hypercubic tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since there were many different naming conventions used by mathematicians re- garding some of the terms studied in this text, this first subsection is devoted to a clarification regarding those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Most mathematicians distinguish among tilings, honeycombs and tessellations, using the last as a generalization of n-dimensional case and the first two to thus refer to tessellations on the plane and on a higher dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Here we employ the same meaning for these terms but avoid their interchangeability which is possible in some sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The definition of a tessellation used here is due to Coxeter [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A tessellation is an infinite set of polytopes fitting together to fill an n-dimensional space with- out overlapping so that the n-1-facet of each polytope belongs to one and only one other polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Tessellations in n − 1 dimensions can be considered as n-dimensional apeiro- topes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' polytopes with infinitely many cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This will be justified in the next section where the connection between polytopes and tessellations will be examined and it will be shown that a polytope may also be regarded as a tes- sellation of a given manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Before this, a few definitions regarding polytopes 3 are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2 Polytope definitions A polytope P is a geometrical figure bounded by finitely many hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It is the n-dimensional generalization of polygon (2-polytope), polyhedron (3- polytope), polychoron (4-polytope), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' with the following properties: The facets of P are polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' P has facets of every dimension 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', n−1 and for brevity a k-dimensional facet is called a k-facet of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Every facet of a facet of P is also a facet of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A convention assumed in this text is the existence of one n-facet of P which is P itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For every two facets F1 and F2 of P, F1 ∩ F2 is also a facet of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For every two facets F1 and F2 of P there exists a uniquely defined facet F1∨F2 of P, namely the smallest facet in terms of inclusion which contains both F1 and F2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The last two properties originate from the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For an n- polytope, each n-1-facet lies in a bounding hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Two or more adjacent bounding hyperplanes intersect at n-2-flat and n-2-facet of P lies in this n-2-flat, three or more adjacent n-2-flats intersect at n-3-flat and n-3-facet lies in this n-3-flat and so on as n or more 1-flats or lines intersect in points which are the vertices of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' There are different naming conventions of the various facets of a polytope besides the obvious for vertex, edge, and face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Examples are cell for 3-facet, hypercell for 4-facet, facet for n-1-facet, ridge for n-2-facet, and peak for n-3-facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For the sake of any further generalization, only the first part of these will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Each constituent which is also a polytope of a lower dimension is additionally referred to as k-face, k-boundary or similar alternative names but here k-facet (k < n) will be used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let Nr denote the number of r-facets of a polytope and Npq denote the number of p-facets that lie in a q-facet if p < q or the number of p-facets that pass through a q-facet if p > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then Nr = Nrr = Nrn and it can be noticed that these numbers are not independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' NpqNq = NqpNp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' They will be called configurational numbers and will be used later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Polytopes may be divided into: convex and non-convex (also known as star- polytopes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' regular, quasi-regular and semi-regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' uniform and non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 4 A set S ⊂ Rn is convex if it has the property that for any pair of points x, y ∈ S, the line segment λx + (1 − λ)y : 0 ≤ λ ≤ 1 with end points x and y, lies entirely in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For any set S, the smallest convex set containing S (the intersection of the family of all convex sets that contain S) is called the convex hull of S, and is denoted by conv(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A convex polytope P is defined to be the convex hull of any finite set of points in Rn [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If F 0 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', F 0 r are the end-points of the edges of P that meet at F 0, then conv(F 0 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', F 0 r ) is an n-1-polytope called the vertex figure of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' An n-polytope P is regular if its facets are regular and its vertex figures are regular (this definition will be revisited in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally, an n-polytope P is uniform if its facets are uniform and its vertex figures are all of the same kind (P is vertex-transitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The dual of a polytope is defined as a polytope whose r-facets correspond to the (n − r − 1)-facets of the original so that its n-1-facets are the vertex figures of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The configurational numbers of the n-1-facet of the dual polytope then naturally correspond to the configurational numbers of the vertex figure of the original polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Star-polytopes will not be considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 2 The regular tessellations and polytopes of n dimensions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 The regular tilings and polyhedra In the following section we will enumerate the regular polytopes in n dimen- sions and will use the same principle to define all regular tessellations of n-1 dimensions as infinite regular n-polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The construction of these is based on Sommerville [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Regular polyhedra satisfy two conditions: (1) their faces are regular polygons of the same kind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' (2) their solid angles are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the classical definition however, the second condition is restated in a stronger form: the vertex figures are regular polygons, which allows for the first condition to be weakened to: their faces are regular polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A more modern definition requires an even stronger condition - the polyhedron must be transitive on its flags, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' face-, edge-, and vertex-transitive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' all faces, edges and vertices must be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The reason for this new definition is that while the requirement for regular faces and vertex figures is sufficient to derive this condition in the usual case, it is no longer valid for the newly introduced abstract polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The two final conditions to be used for the following construction are weaker: (1) each face has the same number n of edges and vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' (2) there is the same 5 number p of edges and faces around each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These two numbers n and p will show to be sufficient for the construction of the same final polytopes as with the stronger conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' From this definition then it follows that N02 = N12 = n and N10 = N20 = p where Nij is the number of i-faces incident to a j-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since each edge is surrounded by two vertices and faces, N01 = N21 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' There- fore using NijNj = NjiNi we arrive at nN2 = 2N1 = pN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' We have to now add one more restriction that will ultimately limit the possible values of n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This restriction is Euler’s polyhedron formula N2 − N1 + N0 = χ where χ denotes a topological invariant called the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A regular polyhedron then has the Euler characteristic of the sphere χ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' To verify this, we need some further tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Gauss - Bonnet formula � M KdA = 2πχ(M) relates the Gaussian curvature K to the Euler characteristic χ as M is any orientable closed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Descartes’ law of closure defect then states that if the polyhedron is homeomorphic to a sphere (hence not necessarily convex), the total angle defect as the sum of the defects of all the vertices is 4π which is the total Gaussian curvature of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Gauss - Bonnet formula then gives 4π = � M KdA = 2πχ(M) ⇒ χ(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The surface of a polyhedron can be considered as a limit case of differentiable surface where the total Gaussian curvature K remains zero at the faces and edges of the polyhedron and is therefore concentrated on the discrete vertex points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Although K is not defined at these points, its integral remains finite and the integral of the total curvature can be replaced with the sum of the angle defect at all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Thus, the number of vertices can be easily derived by dividing 2πχ(M) = 4π with the angle defect at a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N0(2 + 2n p − n)π = 4π The options for the selection of p polygons around a vertex and n vertices around a polygon to get an appropriate angle defect are finally limited to this divisibility criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Although this provides with solutions for p and n, a more geometric approach which can be applied to a higher dimensional case is considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' From Euler’s formula for a regular polyhedron N2 − N1 + N0 = 2 we calculate N0 = 2−N2+N1 = 2− pN0 n + pN0 2 ⇒ N0(1+ p n − p 2) = 2 ⇒ N0 = 4n 2(n + p) − np 6 and analogously for the rest we get N1 = 2np 2(n + p) − np, N2 = 4p 2(n + p) − np, λ = 2 2(n + p) − np which simplifies to N0 = 2nλ, N1 = npλ, and N2 = 2pλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Restricting this to finite regular polyhedra requires finite and positive λ which imposes the additional conditions p > 2, n > 2 and 2(n+p)−np > 0 ⇒ p < 2n n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, for n = 3 ⇒ 2 < p < 6, n = 4 ⇒ 2 < p < 4, n = 5 ⇒ 2 < p < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' n ≥ 6 would imply 2 < p < 3 which is not possible, therefore this depletes all options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' As a result, all possible regular polyhedrons are n 3 3 3 4 5 p 3 4 5 3 3 These are exactly the Platonic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The tetrahedron {3,3} is self-dual, while the hexahedron {4,3} has the octahedron {3,4} as its dual and the icosahedron {3,5} is the dual of the dodecahedron {5,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In a similar way we can take an infinite regular polyhedron by setting 2(n + p) − np = 0 for which λ becomes infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since p > 2 and n > 2, this leads to n 3 4 6 p 6 4 3 These are the three regular tilings, respectively the triangular {3,3}, the hexag- onal {6,3}, and the square {4,4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The case 2(n+p)−np < 0 is a third case but in order to understand it better, we must interpret these results first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the first case, the polyhedron’s configuration of vertices, edges and faces is isomorphic to a tessellation of a sphere where all its vertices lie on the surface and its edges are represented by geodesic arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' To see this, let S denote the total sum of angles of an n-gon on the sphere with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since the total angle deficiency for a sphere is α = � M KdA = 1 r2 A(M) ⇒ A(M) = r2α and for an n-gon 2π − α + S = nπ ⇒ α = (S − (n − 2)π) it follows that the area of the n-gon must be A(M) = (S −(n−2)π)r2 and since �N2 i=1 Si = �N2 i=1 S = 2πN0, and nN2 = 2N1 = pN0, the total area must be ((2N0 − (n − 2)N2)πr2 = (2N0 + 2pN0 n − pN0)πr2 = 2(n + p) − np n N0πr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then for the case 2(n + p) − np = 0 either r, N0 or both have to be infinite in order to have a finite area of the n-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If r is infinite the sphere becomes a 7 plane else N0 must be infinite which makes all vertices indefinitely close to each other on a finite sphere thus again making Euclidean geometry applicable (with sum of angles of n-gon now S = (n − 2)π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The third case follows from the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The last condition 2(n + p) − np < 0 then implies r2 < 0 where a purely imagi- nary radius would require trigonometric formulae which hold on a surface with constant negative Gaussian curvature K called Lobachevski sphere or a hyper- bolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These three cases then help us conclude that regular polyhedrons are equivalent to tessellations of elliptic, Euclidean and hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The above condition 2(n + p) − np > 0 can also be rewritten as 2(n + p) − np > 0 ⇒ 2n p + (2 − n) > 0 ⇒ 2 p > 1 − 2 n ⇒ 1 p + 1 n > 1 2 which is another inequality related to the vertex figure’s angle defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2 The regular cubic honeycomb and polychora In order to continue this construction in n dimensions, we now use the more restrictive classical definition for n-dimensional regular polytope with the two conditions: (1) all n-1-facets of the polytope must be regular polytopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' (2) all vertex figures of the polytope must be regular polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Although it is possible to consider polytopes with n-1-facets that are tessellations of n-2-dimensional space, a restriction only to finite polytopes as n-1-facets and vertex figures is considered for the purpose of this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' To continue the approach from the previous subsection, extend the configura- tional numbers Npq of the polytope with configurational numbers of the n-1-facet Fpq and of the vertex figure Vpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then the following equalities hold because of the definition: Npq = Fpq (p < q < 3), Np3 = Fp (p = 0, 1, 2) Npq = Vp−1,q−1 (p > q > 0), Np0 = Vp−1 (p = 1, 2, 3) The above equalities are true for the specific values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The first equal- ity constitute the relations between the facet and the vertex figure of a 4- polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The second is derived from Np3 = Fp3 = Fp since the n-1-facets of the 3-polytope are 3-facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The third equality makes use of duals N ′: Vpq = F ′ pq = N ′ pq = Np−1,q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally, the last equality is similar to the second in terms of the dual polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, the number of lines or planes through a point F10, F20 in a 3-facet (cell) must be the same as the number of lines or planes V02, V12 through each vertex, hence F10 = F20 = V02 = V12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This is one of the three numbers p, q, r which can finally be extracted from 8 F02 = F12 = N02 = N12 = p F10 = F20 = V02 = V12 = q V10 = V20 = N21 = N31 = r where p, q describe the 3-facet and q, r the vertex figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' They represent the number of vertices or edges of a polygon (p), the number of edges or planes through each vertex of a polyhedron (q), and the number of 3-facets or cells through each edge (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' As mentioned before, by superimposing all available choices for finite polyhe- drons, the following 4-dimensional cases can be constructed: 333 334 335 343 353 433 434 435 533 534 535 However, they still need to be distinguished in terms of metric (elliptic, Eu- clidean, and hyperbolic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' To do this, we have to examine the dihedral angles of the constituent polyhedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The geometric prescription makes use of spherical trigonometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Figure 2: Dihedral angles of a polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let C0 be any vertex a regular polyhedron p,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let C1 be the mid-point of an edge through C0, C2 the center of a face though C0C1, and C3 the center of the polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Denote ∠C1C0C2 = θ1 and ∠C2C1C3 = θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since the polyhedron is regular ∠C0C2C1 = π/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider the unit sphere around C0 - it is cut by the edges and faces through C0 into spherical polygon with q sides (the vertex figure of the polyhedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then θ1 is half the length of the edge of any such polygon and θ2 is half the magnitude of its angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally, since π/q is half the angle which an edge of the polygon subtends at its center, we can use Napier’s rules for a spherical right triangle 9 C1 3b=, B=元/qcos(B) = sin(A)cos(b) → cosπ q = sinθ2cosθ1 = sinθ2sinπ p where 2θ2 is the dihedral angle of the polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The results for the dihedral angles of the regular polyhedrons with a given p and q determine how many polyhedrons can be placed around an edge of the constructed polychoron (4-polytope) which is the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' They are shown in the next table: p q 2θ2 r 3 3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='5o 3, 4, 5 3 4 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='5o 3 4 3 90o 3, 4 3 5 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2o 5 3 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='6o 3 Thus, the only case of polyhedra completely filling a space is the case of 4 cubes around an edge (8 around a vertex) which identifies the case 434 as the only 4-polytope tessellating an Euclidean 3-dimensional space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' the cubic honeycomb which is the only regular honeycomb in 3 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The regular closed polychora which can be viewed as regular tessellations of a hypersphere can only have 3, 4 or 5 tetrahedra, 3 octahedra, 3 or 4 hexahedra (cubes) or 3 dodecahedra at an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' They represent the cases 333, 334, 335, 343, 433, and 533 where 333 and 343 are self-dual, and the others are in dual pairs as 433 and 334 and 335 and 533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Using the two conditions for regularity, the following relations can be obtained N0qrq = N1, N1r = N2p, N2 = N3qqp where qp = 2 2(p + q) − pq , rq = 2 2(q − r) − qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' However, they will not be explained further because Euler’s equation N0 −N1 + N2 − N3 = 0 is homogeneous in four dimensions and can only determine the ratios between these configurational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N0 : N1 : N2 : N3 = pqp : pqqprq : qrpqrq : rrq Therefore, in order to finally determine these polychora, we would have to build them geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since this gets beyond the scope of this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' here they will simply be identified for the reader: {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3} pentachoron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 5-cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' or 4-simplex (analog of the tetrahedron) {4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3} octachoron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 8-cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' or 4-cube (analog of the cube) 10 {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='4} hexadecachoron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 16-cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' or 4-orthoplex (analog of the octahedron) {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3} icositetrachoron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 24-cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' or octaplex (no 3-dimensional analog) {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='5} hexacosichoron or 600-cell (analog of the icosahedron) {5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3} hecatonicosachoron or 120-cell (analog of dodecahedron) One of the algorithms developed for this thesis generates the coordinates of any of these 4-dimensional analogues of the Platonic solids so that various cross- sections can be performed in order to study their symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3 The regular tessellations in n-1 dimensions and n- polytopes As we have seen, the problem of constructing a regular n-polytope is only a part of the general problem of constructing a regular tessellation in n-1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Such a tessellation divides n-1-dimensional space into equivalent n-1-polytopes which in turn are tessellations of elliptic space in n-2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In addition, any hypersphere around a vertex of the n-1-dimensional tessellation will define a tessellation of elliptic space in n-2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Figure 3: Front view of a cube with cases p = 2, q = 0 (Aq-vertex, Ap+1-cube) and p = 2, q = 1 (Aq-edge, Ap+1 − cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider a p+1-facet Ap+1 and a q-facet Aq inside it such that q ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In Ap+1 take any (p + 1 − q)-dimensional flat Sp+1−q intersecting Aq at a point O and construct a small hypersphere centered at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It is easy to see that dimSp+1−q = p + 1 − q = dimAp+1 − dimAq, dimAq ∩ Sp+1−q ≤ min(q, p + 1 − q), and O ⊂ Aq ∩ Sp+1−q ⇒q+r Aq ∩ Sp+1−q = Or, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' all q+r-facets through Aq are cut by Sp+1−q in r-flats through O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These flats then cut the hypersphere in r-1-dimensional regions (the q+1-facets through Aq cut the hypersphere in points, the q+2-facets cut it into great circles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=') 11 S3 A A=0Thus, the figure formed on the hypersphere is a regular tessellation on elliptic space of p − q dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If we denote by aNbc the number of b-facets passing through a c-facet and lying on an a-facet of the general polytope and by aN ′ bc the configurational numbers of the regular tessellation of p − q dimensions, we get aN ′ bc = a+qNb+q,c+q When q = p − 1, a 1-dimensional elliptic regular tessellation can be obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' a regular polygon, and p+1Np,p−1 = 2N ′ 10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' When q = p − 2, a 2- dimensional elliptic regular tessellation is obtained as p+1Np,p−2 = 3N ′ 20 = 3N ′ 10 = p+1Np−1,p−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The numbers p+1Np,p−2 = p+1Np−1,p−2 = kp (p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', n − 1) are analogous to the ones in the 4-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For p = 1, the convention rNp,−1 = Npr is used to denote the total number of p-facets in an r-facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, denote the configurational numbers of the n-1-facet of the tessel- lation by pFqr and the configurational numbers of the vertex figure by pVqr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' p+1Fp,p−2 = p+1Np,p−2 = kp ⇒ k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', kn−2 p+1Vp,p−2 = p+2Np+1,p−1 = kp+1 ⇒ k2, k3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', kn−1 Now, the geometrical method for distinguishing among the elliptic, Euclidean, and hyperbolic case has to be generalized for any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider a regular tessellation k1k2k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='kn of n-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let C0 be any vertex, C1 the midpoint of an edge though C0, C2 the center of a face (polygon) through C0C2 and in general Cr the center of the r-facet through C0C1C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='Cr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then the triangle CpCqCr (p < q < r) is always a right angled triangle with right angle at Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The angle C0C2C1 is half the angle at the center of a plane face subtended by an edge and is equal to π/k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The angle C1C0C2 = θ1 is half the angle between two adjacent edges, C2C1C3 = θ2 is half the dihedral angle between two adjacent plane faces and in general CpCp−1Cp+1 = θp is half the dihedral angle between two adjacent p-facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since there are kn n-1-facets at each n- 2-facet, the angle Cn−1Cn−2Cn = θn−1 = π/kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Now consider a sphere at C0 in the 3-facet C0C1C2C3 which is cut by the lines and planes through C0 in a regular spherical polygon with sides 2θ1 and angles 2θ2 and the polygon subtended at the center of the polygon by half the side is π/k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Hence by spherical trigonometry cosπ/k2 = sinθ2cosθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Again, in the 4-face C0C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='C4 take a hyperplane H perpendicular to C0C1 at C1 and in this hyperplane consider a small sphere centered at C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It is cut by the lines and planes in which H cuts the planes and hyperplanes through C0C1 in a regular spherical polygon with sides 2θ2 and angles 2θ3 and the angle 12 subtended by the center of the polygon by half the side that is π/k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' By spherical trigonometry again cosπ/k3 = sinθ3cosθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally we obtain the formula cosπ/kr = sinθrcosθr−1 (r = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', n − 1) with θn−1 = π/kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' To determine θ1, consider the right triangle C0C1C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In Euclidean geometry the sum of its angles θ1 + π/k1 + π/2 = π and θ1 = π/2 − π/k1 while in elliptic geometry it is more and in hyperbolic less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Now for n = 5 take regular tessellation in 4-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then cosπ/k3 = sinθ3cosθ2 = sinπ/k4cosθ2 and cosπ/k2 = sinθ2cosθ1 sin2θ2 + cos2θ2 = 1 ⇒ cos2θ1sin2θ2 cos2θ1 + sin2π/k4cos2θ2 sin2π/k4 = 1 cos2π/k2 cos2θ1 + cos2π/k3 sin2π/k4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If the last expression is greater than 1, the tessellation will be elliptic, and for less than one it will be hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Using these generalization further for n = 5, we require that the 4-facets and ver- tex figures of a regular 5-polytope must be elliptic tessellations of 4-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since in four dimensions the only such are the six regular polychora 333, 334, 335, 343, 433, 533 the only possible 5-dimensional cases are 3333 3334 3335 3343 3433 4333 4334 4335 5333 5334 5335 Applying the condition cos2π/k2 cos2θ1 + cos2π/k3 sin2π/k4 = 1 we can divide the above cases into three elliptic 3333, 3334, and 4333, three Euclidean 3343, 3433, and 4334 and the other five hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, we can conclude that there are three regular tessellations in an Euclidean space of 4-dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' such space can be completely filled with 8-cells (cubes), 16-cells, or 24-cells and that there are only three regular 5-polytopes - the 5- simplex, the 5-cube, and the 5-orthoplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For n = 6 since the 5-facet k1k2k3k4 and vertex figure k2k3k4k5 must both be elliptic, the tessellations can only be of the form 13 33333, 33334, 43333, 43334 which correspond respectively to 6-simplex, 6-cube, 6-othoplex and finally a reg- ular tessellation of a 5-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It can immediately be seen that in any further dimension the choice will remain the same and the self-dual n-simplex αn, the n-cube γn and its dual n-orthoplex βn are the only regu- lar polytopes in n dimensions (n ≥ 5) while the 43n−24 tessellation δn+1 is the only regular tessellation of an n-dimensional space (notation due to Coxeter [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 3 The quasi-regular and uniform tessellations and polytopes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 Quasi-regular tilings and the crystallographic restric- tion The numbers obtained for each specific regular n-polytope in the previous sec- tion turn out to be very essential for the description of its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' They will reappear in their specific combination later in this text as the orders of the generators for Coxeter systems and in the classification of the edges of Coxeter graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Here we first introduce the most basic notation which was briefly used above called the Schl¨afli symbol and its extension for quasi-regular tessellations first suggested by Coxeter [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For the sake of brevity this extension is analyzed mainly for the case of polyhedra and respectively tilings and a final extension to uniform tessellations then is introduced with an emphasis on honeycombs and polychora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Nevertheless, quasi-regular honeycombs are briefly introduced in the next section for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally, since this subsection will emphasize on tilings, its second part analyses the possible symmetries of all tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Schl¨afli symbol of a polytope with the numbers k1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='kn−1 simply has the form {k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', kn−1} and can therefore be used for all derived regular poly- topes or tessellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, the Schl¨afli symbol of the n-1-facet of an n-polytope {k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', kn−1} is {k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', kn−2}, the Schl¨afli symbol of the ver- tex figure is {k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', kn−1}, and the Schl¨afli symbol of the dual of the polytope is {kn−1, kn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', k1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This notation can now be extended to quasi-regular polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The interior of the intersection of two dual regular polyhedra {p,q} and {q,p} centered at the same point has N1 vertices which are exactly the mid-edge points of both {p,q} and {q,p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Its faces consist of both N0 {q} and {p} polygons which are the vertex figures respectively of {p,q} and {q,p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' There are 4 edges at each vertex and 2N1 edges altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then N0 − N1 + N2 = N1 − 2N1 + (N0 + N2) = 2 14 and the resulting polyhedron can be denoted as � p q � = �q p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The possible cases can be derived based on the above restriction: p = q = 3 ⇒ N ′ 0 = N1 = 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N ′ 1 = 2N1 = 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N ′ 2 = N0 + N2 = 4 + 4 = 8 p = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' q = 4 ⇒ N ′ 0 = N1 = 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N ′ 1 = 2N1 = 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N ′ 2 = N0 + N2 = 6 + 8 = 14 p = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' q = 5 ⇒ N ′ 0 = N1 = 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N ′ 1 = 2N1 = 60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' N ′ 2 = N0 + N2 = 12 + 20 = 32 For the three elliptic cases we then get �3 3 � = {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 4}−octahedron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' �3 4 � −cuboctahedron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' and �3 5 � −icosidodecahedron and since the edges are all alike,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' each separating p from q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' this gives rise to the definition of a quasi-regular polyhedron as a polyhedron with exactly two kinds of regular faces that is edge-transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The proof that these are the only quasi-regular polyhedra originates from the fact that the dihedral angles at a vertex make a total that must conform to the inequality r(1 − 2 p)π + r(1 − 2 q )π < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, 1 − 2 p + 1 − 2 q < 2 r ⇒ 1 − 1 p − 1 q < 1 r ⇒ 1 p + 1 q + 1 r > 1 and since p and q cannot be less than 3, r = 2 and p = 3, q = 4 or p = 3, q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In addition we can consider the dual pair of regular tilings {3,6} and {6,3} where the trihexagonal tiling �3 6 � is obtained with vertices as the intersections of the their edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The symmetry group of such a tiling is an infinite group of congruent trans- formations in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This group contains a finite subgroup of index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A classification of the symmetries of the plane tilings then leads to the crystallo- graphic restriction theorem which states that if a discrete group of rotations in the plane has more than one center of rotation, then the only rotations that can occur are of order 1, 2, 3, 4, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider the Euclidean motion group R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='O(2) of isometries on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Any finite subgroup of this group fixes a point and so is conjugate to a finite subgroup of O(2) that fixes the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The finite subgroups of O(2) are then the cyclic groups of order n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' rotations of 2π/n, and the dihedral groups of order 2n with rotations as a subgroup of index 2 and reflections that conjugate those to their inverse rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The subgroup R2 consists of all translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider all discrete cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' rotations and translations that cannot be arbitrarily close to the identity transformation and are bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let Γ ≤ O(2) be 15 discrete subgroup and consider the lattice group L = Γ∩R2 = Za+Zb generated from translations by two linearly independent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, let ¯Γ be the image of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='O(2) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='O(2)/R2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' ¯Γ = Γ/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then ¯Γ preserves the lattice L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The proof is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Take a vector b ∈ L or equivalently a translation by this vector tb ∈ Γ and take γ ∈ Γ that maps to ¯γ ∈ ¯Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In terms of linear operators set γ(v) = Av, tb(v) = v + b, γ−1(v) = A−1v γtbγ−1(v) = γtb(A−1v) = γ(A−1v + b) = A(A−1v + b) = v + A(b) Therefore γtbγ−1 = t¯γ(b) is a conjugate translation by ¯γ(b) and since γ, tb ∈ Γ by the operation closure it follows that t¯γ(b) ∈ Γ and finally ¯γ(b) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Because of this established fact, ¯Γ = Cn or ¯Γ = D2n with n = 1, 2, 3, 4, 6 and ¯Γ = Cn as rotation parts has maximum order of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A proof in terms of linear operators is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let A ∈ ¯Γ be a rotation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' detA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' We have to show that the order of A is 1,2,3,4 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider the characteristic polynomial of A x2 − tr(A)x + det(A) = x2 − tx + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider A to be a rotation by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then since rotation is analogical to scaling with complex numbers, x2 − tx + 1 has complex roots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' t2 − 4 ≤ 0 (the only real cases are +1 and −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Thus, the matrix is diagonalizable over the complex numbers and tr(A) = t = z + ¯z is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, since A stabilizes the lattice L = Za + Zb, it takes both a and b to integer multiples of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If we take these as column vectors of A in the basis a, b, A will have integer entries in this basis and thus its trace t is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then t2 − 4 ≤ 0 ⇒ t = ±1, ±2, 0 where t = ±2 = 2cos(θ) are the rotations of order 1 and 2, t = ±1 = 2cos(θ) are the rotations of order 3 and 6, and t = 0 = 2cos(θ) are the rotations of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Considering Gaussian integers for coordinates, the symmetry group of the tiling {4,4} is then generated by the translation z′ = z + 1 and the rotation z′ = iz of order 4 while the symmetry group of the {3,6} is generated by the same translation along with a rotation z′ = eπi/3z = (e2πi/3 + 1)z of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The crystallographic restriction theorem can be used to classify all symmetries of a tiling on the plane with translations, rotations, reflections, and glide reflections as isometries of the Euclidean plane, also called wallpaper groups or crystallo- graphic groups on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2 Uniform honeycombs and geometric operations The extension of the Schl¨afli symbol introduced by Coxeter can be generalized for rectified n-polytopes where rectification is the process of taking the inter- section of two dual polytopes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' cutting the vertices at the midpoints of the edges which will result in a polytope bounded by both the vertex figures and rectified faces of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 16 For a quasi-regular honeycomb, all cells must be regular and all vertex figures must be quasi-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Alternative conditions then are that the vertex figures are all the same and the cells are of two alternating kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The only two regular polyhedra, whose angle sum divides 2π are the octahedron and the tetrahedron (their sum is π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The only quasi-regular honeycomb then is �3, 3 4 � or the al- ternated cubic honeycomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It can be seen as a cubic honeycomb with alternate vertices removed reducing cubic cells to tetrahedra and creating octahedron cells in the remaining gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Rectification is a special case of the more general truncation operation which can be used to derive a list of uniform polytopes and tessellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Although there is a new notation known as the Wythoff symbol originating from the Wythoffian construction of uniform polytopes, there is also a further extension of the Schl¨afli symbol and a notation that will be introduced in the next section that adds more information about the symmetries of the tessellation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' of the polytope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The final extended Schl¨afli symbol denotes the k-th rectifica- tion of a polytope as tk{p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', pn−1} where t0,1 is a truncation applied to polygons or polytopes of higher dimension, t0,2 is cantellation (both edges and vertices removed) applied to polyhedrons or higher, t0,3 is runcination, t0,1,2 is cantitruncation (cantellation and trucation), t0,1,2,3 is runcicantitruncation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' All these geometric operations can be generalized in terms of sequences of cross- sections with higher dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Truncation results from the sequence of cross-sections parallel to a facet of the vertex figure of the polytope where crossing the vertex leads to intersection of the previously crossed edges and origination of the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Cantellation is analogously the result of sequence of cross-sections parallel to an edge of the polytope where crossing the edge leads to intersection of previously crossed faces and origination of the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Such a generalization can be used for the reconstruction of an n + 1-dimensional poly- tope through the geometrical operations performed on its cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Using this, we were able to reconstruct the graph of a 4-polytope with cross- sections which are the gradual truncations of a cube to its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These operations (besides rectification) introduce the semi-regular polytopes in addition to the quasi-regular and regular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' All of them are unified under the definition of a uniform polytope whose only two conditions are: (1) uniform polytope facets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' (2) vertices of the same kind (vertex-transitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The uniform 2-polytopes are necessarily the regular polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' There are 11 uniform tilings of the plane and 28 uniform convex honeycombs in 3 dimensions, also called the Archimedean honeycombs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' From the latter, there are just one regular (cubes) and one quasi-regular (octahedra and tetrahedra), both mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Truncation has been used to derive 7 additional hon- eycombs originating from the cubic one and 4 additional originating from the 17 alternated cubic honeycomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally there are 15 more from prismatic forms derived from modifications of the uniform plane tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 4 Coxeter groups and reflection groups 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 Coxeter systems In the previous section it was shown that the group of isometries on the Eu- clidean plane admits only a few discrete subgroups of O(2) thus allowing a classification of the possible symmetries of a tiling that is not necessarily uni- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The 3-dimensional case of crystallographic groups is the space groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A formal description of the symmetries of a tessellation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' polytope) is the goal of this section which leaves the techniques of n-dimensional geometry aside and uses tools mainly from abstract algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A Coxeter system is a pair (W, S) where W is a group and S a set of generators S ⊂ W restricted by the relations (sisj)m(si,sj) = 1 where m(si, si) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' si is an involution, and m(si, sj) = m(sj, si) ≥ 2, ∀si ̸= sj ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If there is no relation between si and sj the convention m(si, sj) = ∞ is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The group W is then the quotient F/N where F is a free group on the set S and N is the normal subgroup generated by all elements (sisj)m(si,sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, |S| = n is the rank of the Coxeter system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' W(M) is then a Coxeter group that can be constructed from a symmetric n × n matrix M = (mij)1≤i,j≤n indexed by S with entries in Z ∪ {∞} such that mii = 1 and mij ≥ 2 ∀i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Coxeter group of type M is then analogously W(M) = ⟨s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' ∈ S|(sisj)mij = 1, mij ∈ M⟩ which will be denoted W when no ambiguity is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A few lowest rank examples are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If |S| = n = 1 then M = (1) and W(M) = ⟨s1|s2 1 = 1⟩ which is the cyclic group of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For n = 2 then M = � 1 m m 1 � ⇒ W(M) = ⟨s1, s2|s2 1 = 1, s2 2 = 1, (s1s2)m = 1⟩ = D2m for m ∈ N∪∞ which is the Klein Four group for m = 2, the dihedral group of finite order for 2 ≤ m ≤ ∞ and the infinite dihedral group D∞ for m = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Instead of the matrix, the Coxeter system (W,S) can be constructed from an undirected graph Γ with a vertex set S where two vertices s and s′ are joined with an edge that is labeled m(s, s′) if 3 ≤ m ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore if the distinct ver- tices s and s′ are not joined, then m(s, s′) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The edges with label m(s, s′) = 3 are omitted due to their frequency and by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The resulting graph Γ is called Coxeter graph and as a notation contains more information than the extended Schl¨afli symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 18 Figure 4: Sample Coxeter graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The entire information about the matrix can be reconstructed from the Coxeter graph as the same applies to the Schl¨afli symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' To show this we need to study the connection of the Coxeter groups to the reflection groups and the groups of symmetries of the regular polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Because Coxeter groups are based on an abstract presentation, they do not necessarily admit a faithful representation as reflection groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The abstract group of each reflection group is a Coxeter group as the reflections are a special case of involutions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Infinite Coxeter groups in particular may not admit a representation as a reflection group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' However, finite Coxeter groups have a faith- ful linear representation as groups generated by reflections in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Some of the finite linear groups generated by such reflections are groups of sym- metries of the regular polytopes in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' We will show that Coxeter groups admit a representation since this is essential for the purpose of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let (W, S) be a Coxeter system of type M and take n = |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' We will now construct a real linear representation of W of degree n such that the images of the elements of S are reflections in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A reflection on a real vector space V is a linear transformation on V fixing a subspace of V of codimension 1, a reflection hyperplane Hα, and having a nontrivial eigenvector α with eigenvalue λ = −1, called a root of the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Now take the vector space V over the field R with a basis {αs|s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Define a bilinear form B on V such that B(αs, αs′) = −cos π m(s, s′) where m(s, s′) = mij ∈ M and since mij = mji the form is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then B = −1 for m(s, s′) = ∞, B(αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='αs) = 1, and B(αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='αs) ≤ 0 for s ̸= s′ with equality only for m(s, s′) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Now for each s ∈ S define a reflection σs : V → V such that σsλ = λ − 2B(αs, λ)αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then σsαs = −αs and σsλ = λ with {λ ∈ V |B(x, λ) = 0} := Hs the hyperplane orthogonal to αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore σs has order 2 in GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Additional observation from the definition is that B(σsλ, σsµ) = B(λ, µ) ∀λ, µ ∈ V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' the reflection preserves the bilinear form and each element generated by σs will preserve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A final observation is that |σsσs′| = m(s, s′) and should be proven next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider Vs,s′ := Rαs⊕Rαs′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The restriction of B to Vs,s′ is positive semidefinite 19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 1 2 n-1 nsince for λ = aαs + bαs′ we get B(λ, λ) = B(aαs + bαs′, aαs + bαs′) = = a2B(αs, αs) + 2abB(αs, αs′) + b2B(αs′, αs′) = = a2 − 2abcos(π/m(s, s′)) + b2 = = a2 − 2abcos(π/m(s, s′)) + b2(cos2(π/m(s, s′)) + sin2(π/m(s, s′)) = = (a − bcos(π/m(s, s′)))2 + b2sin2(π/m(s, s′)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore the form is positive definite if sin(π/m(s, s′)) ̸= 0 and m < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Note further that σs and σs′ stabilize Vs,s′, so the order of σsσs′ as an operator in Vs,s′ has two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' (1) If m < ∞ since the form is positive definite we can consider Euclidean plane and since B(αs, αs′) = −cos(π/m(s, s′)) = cos(π−(π/m(s, s′))) and the angle between Rαs and Rαs′ is therefore π −(π/m(s, s′)), the angle be- tween Hα and Hα′ is π/m(s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Because rotation through 2π/m(s, s′) can be achieved as a product between two reflections with an angle between their fixed hyperplanes π/m(s, s′), it follows that σsσs′ has order m(s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The fact that B is positive definite on Vs,s′ implies that V = Vs,s′ ⊕ V ⊥ s,s′ where V ⊥ s,s′ := s⊥ ∩ s′⊥ is fixed by both σs and σs′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then σsσs′ has order m(s, s′) also on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' (2) If m = ∞, B(αs, αs′) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then if λ = αs + αs′, B(λ, αs) = B(λ, αs′) = 0 and σs and σs′ fix λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then σsσs′αs = σs(αs + 2αs′) = 3αs + 2αs′ = 2λ + αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Applying σsσs′ k ∈ Z times, (σsσs′)kαs = 2kλ + αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore we can conclude that σsσs′ has infinite order on Vs,s′ and therefore also on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This helps us conclude that there is a unique homomorphism σ : W → GL(V ) where σ(s1s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='sn) = σ(w) = σw = σs1σs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='σsn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' We call this homomorphism a linear representation of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The fact that this representation is then faithful is a corollary from another theorem regarding the length function for the Coxeter group and is outside of the scope of this text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' However, one final definition which is necessary for the later sections is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The subgroup WI generated by a given subset I ⊂ S and any of its conjugates are called parabolic subgroups of the Coxeter group W and are in fact Coxeter groups themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2 Root systems Consider a Coxeter system (W, S) of type M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The set Φ := � s∈S,w∈W σ(w)αs is called the root system of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It consists of the collection of orbits of the unit vectors in the reflection representation space V on which W acts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' the collection of unit vectors in V permuted by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These remain unit vectors because W preserves the bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, Φ = Φ+ ∪ Φ− where Φ+ := Φ ∩ ( � s∈S R≥0αs), Φ− := Φ ∩ ( � s∈S R≤0αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A root α is called positive (write α > 0) if α ∈ Φ+ and negative (α < 0) if α ∈ Φ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Note that Φ = −Φ since σs(αs) = −αs, Φ ∩ Rα = {α, −α} ∀α ∈ Φ, and σ(W)Φ = Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Positive and negative roots are defined with regard to certain 20 total ordering like lexicographic ordering where � aiλi < � biλi ⇒ ak < bk where k is the least index such that ak ̸= bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A subset ∆ ⊂ Φ+ of vectors αi constituting a basis for the R-span of Φ in V is called a simple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A root α ∈ ∆ is called simple root of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Each α′ ∈ Φ is a linear combination of ∆ with coefficients all of the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Further relation between the roots and the reflections can be established through the consideration of the set R = {wsw−1|w ∈ W, s ∈ S} or reflections of the Coxeter system (W, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' By the geometric representation σ : W → GL(V ), each s ∈ S acts on V as a reflection σ(s)αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' More generally, a reflection in GL(V ) can be associated to each root α ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider α = σ(w)αs := w(αs) (for brevity) for w ∈ W and s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then wsw−1 ∈ R acts on W as follows: σ(wsw−1)λ =: wsw−1(λ) = w[w−1(λ) − 2B(w−1(λ), αs)αs] = = λ − 2B(w−1(λ), αs)w(αs) = λ − 2B(λ, w(αs))w(αs) = = λ − 2B(λ, α)α which shows that wsw−1 does not depend on the choice of w and s but only on the choice of α so can be denoted sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, sα acts on V as a reflection sending α to −α, fixing pointwise the hyperplane Hα ⊥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' As a consequence, α and −α both determine the same reflection sα = −sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The root-reflection corre- spondence is finally established due to the bijective map α → sα (for α ∈ Φ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, each reflection of a Coxeter group W has a unique positive root α ∈ Φ+ and each α ∈ Φ+ is the positive root of a unique orthogonal reflection with respect to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If α, β ∈ Φ and w, w′ ∈ W such that w′(β) = w(α), then wsαw−1 = w′sβw′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This correspondence helps us interpret the relations in a Coxeter system of the form (sisj)mij as the result of two reflections fixing hyperplanes meeting at an angle π/mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The element sisj ∈ S being of order mij than has the geometrical interpretation of a rotation by 2π/mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Some further important properties and definitions of root systems to be men- tioned here are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If ∆ is a simple system in Φ, then (α, β) ≤ 0 ∀α ̸= β ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If αs ∈ ∆ then s(Π\\{αs}) = Π\\{αs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A root system Φ is crystallographic if it satisfies the additional requirement 2(α, β) (β, β) ∈ Z ∀α, β ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The vector α∨ := 2α/(α, α) is called a coroot of α ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The set Φ∨ of all coroots is the dual root system or the inverse root system of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The reflections w ∈ W generated by Φ∨ are the same as Φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' wα∨ = w(α)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A final definition for this section should be given for the fundamental domain of the action of W on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Take a positive system Φ+ containing a simple system ∆ 21 and consider the open half-spaces H+ α := {λ ∈ V |(λ, α) > 0} and H− α := −H+ α of each reflecting hyperplane Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Define C := ∩αs∈∆H+ α which is open and convex as intersection of open and convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let D := ¯C be the closure of C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' the intersection of the closed half-spaces H+ α ∪ Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Then D = {λ ∈ V |(λ, α) ≥ 0 ∀α ∈ ∆} and each λ ∈ V is conjugate to exactly one point in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Thus D is called a fundamental domain for the action of W in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This could offer a different insight into the geometric representation of the Cox- eter group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The nodes of the Coxeter graph represent the walls of the fundamen- tal domain and two nodes are joined by a branch whenever the corresponding walls are not perpendicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Moreover, the branches are marked with numbers mij > 2 to indicate the angles π/mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the case of a connected graph with- out any even marked branches, all the reflections in the group are conjugate to one another [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This interpretation of the Coxeter graphs in terms of fun- damental domains originally proposed by Coxeter leads to their final extension to the geometrical operations discussed in the previous section and the uniform tessellations and polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='3 Classification All symmetry groups of regular polytopes are finite Coxeter groups (and resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' finite reflection groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' All symmetry groups of regular tessellations are affine Coxeter groups (and resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' affine reflection groups containing normal abelian subgroups such that the quotient group is finite and is itself a Coxeter group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Coxeter graph of an affine Coxeter group is obtained by adding an addi- tional vertex as in the construction performed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The dual polytopes or tessellations have the same symmetry groups and therefore the same Coxeter groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For the regular polytopes in any number of dimensions, the symmetry groups are respectively the symmetric group Sn+1 or the Coxeter group of type An for the regular n-simplex αn, and the hyperoctahedral group or the Coxeter group of type Bn = Cn for the n-cube γn and its dual the n-orthoplex βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The root systems for these use the same notation with the difference that Bn and Cn are interchanged (dual root systems with B2 and C2 isomorphic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The root system Bn of γn which is specifically important for one of the algorithms described in the next section is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Let V = Rn, and let Φ consist of all integer vectors in V of length 1 (short roots) or √ 2 (long roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The total number of roots is 2n2 with 2n short roots ±ei and 2n(n − 1) long roots ±ei ± ej(i < j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For ∆ take the simple long roots αi = ei − ei+1, for 1 ≤ i ≤ n − 1, and the short root αn = en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The reflection s through the hyperplane perpendicular to the short root αn is then the negation 22 of the n-th coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, the symmetry group of the pentagon is H2, the symmetry group of the dodecahedron and its dual icosahedron is the full icosahedral group H3 and the symmetry group of their 4-dimensional analogues (the 120-cell and the 600-cell) is H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The symmetry group of the 24-cell is F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Coxeter groups of type Dn (n-demihypercube), E6 (221, 122), E7 (321, 231, 132), and E8 (421, 241, 142) are the symmetry groups of certain semiregular polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The symmetry group of the hexagon is G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The affine Coxeter groups are then classified as ˜ An for the simplectic uniform tessellation, ˜ Bn for the demihypercubic uniform tessellation, ˜ Cn for the hyper- cubic uniform tessellation, ˜ E6 for 222, ˜ E7 for 331, 133, ˜ E8 for 521, 251, 152, ˜F4 for 16-cell and 24-cell uniform tessellations and ˜ G2 for hexagonal and triangular tiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The final notation to be reviewed in this work is the ringed Coxeter graph which contains enough information to explicitly enumerate almost all types of uniform polytopes and uniform tessellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Each uniform polytope can be generated using the mirror hyperplanes bounding the fundamental region and a single generator point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The reflections of the point through the mirror hy- perplanes and their further reflections through the same hyperplanes form the set of vertices of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The edges of the polytope connect each point to a mirror point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' the faces can be constructed as cycles of edges, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The location of the generating vertex is specified as all nodes of the Coxeter graph corresponding to the mirror hyperplanes on which the vertex does not lie are marked with a ring (equidistant from all ringed node hyperplanes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Thus, all mirror hyperplanes where the generating vertex lies do not generate additional vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A diagram needs at least one active node to represent a polytope and therefore all Coxeter graphs of the regular polytopes have their first node ringed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The more general case of uniform polytopes with one marked hyperplane cor- responds to a generating point at a vertex of the fundamental domain (which is always a simplex because of the way it is constructed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If all nodes are ringed, the generator point lies in the interior of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Generally, if n nodes of the Coxeter graph are marked, the generating point gets n−1 degrees of freedom on n-1-facets of the fundamental domain and can generally be taken to be at the center the n-1-facet for equal n-1-faces of the final polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A secondary fea- ture can be used for the special cases of uniform polytopes with non-reflectional symmetry where the central dot of a ringed node is removed to imply alternate nodes deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The constructed polytope will then only have a subsymmetry of the original Coxeter group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Eventually if all nodes are marked in this way, the polytope is called a snub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Using this final notation we can describe for exam- ple the cuboctahedron, rhombicuboctahedron, octahedron, truncated cube and other uniform polytopes derived from geometric operations on the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 23 5 The hypercube tessellation and convex poly- tope algorithms 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='1 The developed algorithms Two of the algorithms developed for this work will be presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Since they consist of some common approaches and differ by removable compo- nents, they will be described in terms of one sequentially consistent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Although our initial plan was to use parametric equations, it offered no good solution when it comes to the calculation of the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Furthermore, al- gebraic equations did not allow for plotting of vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The hypercube tessellation algorithm generates an n-cube and n additional n- cubes on all of its sides and obtains a cross-section with a 3-dimensional space parallel to three independent vectors (preferably roots from the Bn root sys- tem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This results in the 3-dimensional plot of a few space-filling polyhedra that build up the tessellation and an additional optional 2-dimensional plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The convex polytope algorithm generates a convex n-polytope by analyzing its possible k-facets from its vertex coordinates and then performs a similar cross- section through specified parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For an n-dimensional convex polytope with V as the set of its vertices and F as the set of its n-facets, take all subsets S ⊂ V, |S| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For these n points A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', An find a nonzero vector (u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', un) such that it is orthogonal to the vectors Ai − An, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Thus, we need to solve a homogeneous system of n − 1 equations in n variables u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The nullspace of the con- structed matrix can be more than one dimension in case the n vertices constitute a subspace of the hyperplane (there at least two linearly independent non-zero vectors in the kernel of the considered linear map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These cases immediately imply that the given S is not an n-1-facet of the polytope and therefore can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The remaining results are the normal vectors ⃗nj of the hyperplanes Hj passing through the points from S where j indexes each of the remaining cases for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It is important to consider both positive and negative orientation of the ⃗nj when Hj is tested for an n-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For this purpose one of the definitions of convexity for the polytope has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A polytope is convex if it lies entirely on one side of each of its n-1-facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, if the polytope is convex and has a facet lying in the hyperplane Hj, all points A ∈ V \\S must lie in the closed negative half-space H− j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The signed distance is obtained from the projection of any vector from the hyperplane to the 24 point A = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', an) onto the normal vector of the plane ⃗nj = (u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' |(⃗a − ⃗x) · ⃗nj| | ⃗nj| = |a1u1 + a2u2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' + anun − u1x1 − u2x2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' − unxn| | ⃗nj| = = |a1u1 + a2u2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' + anun − (−C)| | ⃗nj| = a1u1 + a2u2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' + anun + C � u2 1 + u2 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' + u2n The translation constant C can be obtained from the regular hyperplane equa- tion of Hj evaluated at any point A ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The convex hull defined by the vertices in V is then � H− jk = conv(V ), k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=', |F| which is the desired polytope and the solution of |F| inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' However, only the normal vectors njk of the n-facets of the polytope are nec- essary for the second part of the algorithm which is the cross-section with a 3-subspace where we can observe certain symmetries of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Three linearly independent vectors and a translation point are sufficient for defining a unique 3-subspace inside the n-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In order to observe symmetries of the polytope in the 3-subspace, only specific orientations are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' These are determined by the root system Φ of the Coxeter group for the specific polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Thus, picking a 3-subspace which is parallel to three linearly independent roots (not necessarily simple) ensures that the observed cross-section conforms to the symmetries of a parabolic subgroup of the original Coxeter group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It then can in term be described with the root system stabilized by this subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For example considering the root system B4 and a hypercube, we can pick the long roots u1 = (-1,-1,0,0) ∠(u1, u2) = π/3 u2 = (-1,0,-1,0) ∠(u2, u3) = π/4 u3 = (-1,0,0,0) ∠(u3, u1) = π/4 which will result in B3 root system and the cross-section with the hypercube should be invariant under all reflections along these roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The example cross- section is a cube standing on its vertex with respect to the xy-plane and the resulting 2-dimensional cross-section is a hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, instead of picking any possible orientation of the 3-subspace which is also possible, we pick only vectors α ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It is important to note that the choice of a normal vector of the 3-subspace is not unique if its codimension in the n-space is greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For a larger codi- mension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' larger dimension of the orthogonal complement of the 3-subspace, we would have to select a basis of n−3 vectors in order to uniquely identify the 3-subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Therefore, a much better approach is to simply select three vec- tors spanning the 3-subspace in order to determine its orientation in the n-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 25 After the three roots are selected, the Gram-Schmidt orthogonalization process can be used to produce the new orthonormal basis and a transition matrix T with the unit vectors as columns vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Once the linearly independent roots are chosen, orthogonalization inside the 3-subspace naturally does not influ- ence the cross-section in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The n-facet normal vectors T −1njk = T tnjk then define new positive half-spaces which reorients the convex polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the case of the hypercube tessellation algorithm, the same transformation is applied to all side-cubes with the difference that they are initially translated along some initial unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' When the half-space inequalities are produced, translation of the hyperplanes is applied using the point specifying the trans- lation of the 3-subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Finally, the inequalities are solved for x1, x2, and x3 given xi = 0, 3 < i ≤ n and the result is ready for plotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='2 Obtained results The following subsection introduces some important observations and results obtained in this work from both the algorithms and the performed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider the vector subspaces K ⊂ L ⊂ M where dimK = k, dimL = l, and dimM = m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Take a sequence of m linearly independent roots from Φ which uniquely determine M, such that the first l of them uniquely determine L and the first k of them uniquely determine K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The permutation of the m roots leads to different orientation of the cross-section with the n-polytope in M and to different selection or permutation of the k roots for K and of the l roots for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Consider the permutations of m roots that stabilize the k roots in K but do not stabilize the l roots in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' As a result the cross-sections of the polytope in L are changing but the cross-sections with K remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' This provides with a good example of multiple polytopes in l dimensions that have the same k cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' For this same reason, a cross-section of a plane through the origin of a specifically oriented octahedron or cube both result in a hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the same way the observed polytope in the 3-subspace remains unchanged while many different orientations of the original polytope and cross-section polytopes can be observed in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Another result of even higher importance is related to hidden symmetries of the n-polytope that could not be obtained through its root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Choosing three independent roots guarantees that the cross-section will pertain at least the symmetries resulting from these roots i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' will be invariant under reflection through the hyperplanes orthogonal to the roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' However, the cross-section might have additional symmetries and could remain invariant under additional reflections that do not preserve the original n-polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' We can easily illustrate this with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A cross-section of 3-cube with a 2-flat that intersects the middle points of six of its edges results in a hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The symmetry that should formally be observed 26 is the one of the D6 group since all roots of the cube that are parallel to the 2-flat are long and generate A2 root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The angle between two roots parallel to the 2-flat is 2π/3 and thus reflection along their respective hyperplanes (planes in this case) result in a 3-fold rotation that will preserve the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Another cross-section that is parallel to these two roots and intersects three vertices of the original cube gives an equilateral triangle where the extra symmetry is already not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' In the case of the initial cross-section, the roots of the A2 root system all point to the vertices of the hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' As an even-sided polygon, the hexagon is also invariant under reflection defined by another root which points to the middle of its side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Figure 5: Hidden symmetry in the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Denote the two simple roots of the G2 root system α (pointing to a vertex and coinciding with a root from the required A2 root system) and β (point- ing to the middle of a side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' After observing the net of the cube, it can eas- ily be concluded that all roots of the hexagon of type β point to coordinates (k/3, l/3), k = 1, 2, l = 1, 2 on a given face of the cube, a total of four for each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A general reflection of the cube along the β vector fixes the orthogonal plane Hβ crossing two opposite vertices and the middle points of two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' It does not preserve the cube and therefore does not belong to the symmetry group of the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' As a result the β vector cannot be a root of the B3 root system and the D12 group as the symmetry group of the hexagon is not a subgroup of the octahedral group Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Parabolic subgroups of the Coxeter group of a hypercube thus do not encompass all symmetries that might be observed in a cross-section with the hypercube and selecting roots from its root system only ensures the minimum symmetry of the particular cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Symmetry of larger order can be obtained through cross-sections of higher di- mensions in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A central cross-section with a 4-cube that is parallel to three independent long roots generating a root system A3 results in a specif- ically oriented octahedron (instead of a tetrahedron) which remains invariant under reflections along all roots in A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Hidden symmetries are also the rea- 27 son why uniform polyhedra like the cuboctahedron cannot directly be obtained from intersections with an n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Cuboctahedron can easily be obtained from a central cross-section with a 16-cell (4-orthoplex) that is parallel to three short roots of the Cn root system and 16-cell can be obtained from a 4-dimensional central cross-section with the 5-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' However, any further permutation of the sequence of four roots which are sufficient for a 16-cell cross-section will result in a �4 3 � possible 3-dimensional cross-sections none of which will take advantage of the additional symmetry of the 4-dimensional cross-section necessary for ob- taining the cuboctahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' A final conclusion then concerns the 3-dimensional cross-sections that can be obtained from an n-dimensional cubic tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Rotating the tessellation will result in different cross-sections with its space-filling hypercubes and therefore different additional symmetry of the resulting space-filling polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Returning to the simple 3-dimensional example, the squares tiling can be obtained from a face-first intersection with the cubic honeycomb and the equilateral triangles from a vertex-first cross-section through three of the vertices of any of the cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The hexagon tiling however can be achieved only if the cubes are interpreted as polyhedra with additional vertices at the center of each original face with solid angles of π and the original cubes are then arranged in a way so that the vertex of a cube touches the center of the original face of another and its edges remain parallel to the edges of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' If the original cubic honeycomb is retained and at least one of the cubes is intersected at the centers of six of its edges, the resulting tessellation will be the trihexagonal tiling of the plane which consists of polygons with additional symmetry and polygons with the minimum necessary symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 28 Appendices A Visualizations Figure 6: The convex polytope algorithm - rhombic dodecahedron obtained from 24-cell cross-section and hexagonal bipyramid obtained from 16-cell cross- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Figure 7: The hypercube tessellation algorithm - trihexagonal tiling and alter- nated cubic honeycomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 29 References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Gruenbaum and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Shephard, Convex Polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Bulletin London Mathematical Society, Oxford, 1 (1969), 257-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Sommerville, An Introduction to the Geometry of N Dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Dover Publications, New York, 1st Edition, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Coxeter, Regular Polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' The Macmillan Company, New York, 2nd Edition, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' [4] James E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Humphreys, Reflection Groups and Coxeter Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1st Edition, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} +page_content=' 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQf6v6W/content/2301.01880v1.pdf'} diff --git a/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf b/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0ee702afa285c59651d5fc55f1e58de57410dc5 --- /dev/null +++ b/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4e3b00b7402fcec90ffa426ff7e00d8e40283ce908c5a137853e9ab8bb5c091b +size 1287801 diff --git a/tNAzT4oBgHgl3EQfBfpr/vector_store/index.pkl b/tNAzT4oBgHgl3EQfBfpr/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..41dc752ae014c75a97448a011507ac4a287f5a7f --- /dev/null +++ b/tNAzT4oBgHgl3EQfBfpr/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7dc62f6a59dad7babc02c9d1c9a89ac287035f1cccab301511ef4f0fb7ea42cb +size 436597 diff --git a/ttFJT4oBgHgl3EQfdiwx/content/tmp_files/2301.11548v1.pdf.txt b/ttFJT4oBgHgl3EQfdiwx/content/tmp_files/2301.11548v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b183b5d17db0b43dcace1083cdc4fa8df444dae --- /dev/null +++ b/ttFJT4oBgHgl3EQfdiwx/content/tmp_files/2301.11548v1.pdf.txt @@ -0,0 +1,863 @@ +arXiv:2301.11548v1 [quant-ph] 27 Jan 2023 +No-signaling in Nonlinear Extensions of Quantum Mechanics +Rohit Kishan Ray∗ +Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal - 721302, India +Gian Paolo Beretta† +Department of Mechanical and Industrial Engineering, University of Brescia, 25123 Brescia, Italy +(Dated: January 30, 2023) +Devising a nonlinear extension of quantum mechanics is nontrivial because unphysical features +such as supraluminal communication (signaling) are to be excluded. In this Letter, we show that +the steepest entropy ascent formalism is a viable no-signaling extension belonging to a broader class +of no-signaling nonlinear evolution equations for which the local evolution of a subsystem is not +necessarily bound to depend only on its reduced state. +Quantum mechanics (QM) in its most common form +(Schr¨odinger–von Neumann formalism) is linear in state +space, where linear operators operate upon state vectors, +and the time evolution is linear. In 1989, when the late +Prof. Weinberg asked to test the linearity of quantum +mechanics as we know it [1], a new quest began for the +physicists working in the field. Thence onward, distin- +guished researchers such as Gisin [2] and Polchinski [3] +showed that introducing non-linearity via operators pro- +duces signaling (faster than light communication), lead- +ing to the violation of causality. +In this context as a +consequence of using linear operators in QM, we must +take note of the fact that no-cloning, first introduced by +Park [4] and later explicitly shown by Wootters, Zurek, +and Dieks [5], is sufficient for no-signaling. Nonlinear- +ity, when introduced via stochastic QM through Lind- +blad operator formalism [6] for open quantum systems, +has been shown to respect no-signaling. Ensuing work by +Ferrero et al. [7] has shown that nonlinearity in QM can +be accommodated without compromising no-signaling if +we consider the time evolution to be nonlinear albeit +maintaining linearity in the state space and operators. +In a more recent work [8], it has been shown that convex +quasilinear maps can contribute to the nonlinear dynam- +ics of QM without invoking signaling, and this formalism +retains much of QM as it is. Thusly, Rembieli´nski and +Caban [8] have found a minimal permissible deviation +from the linear structure of QM dynamics. +For about four decades, a nonlinear extension of quan- +tum mechanics initiated by Hatsopoulos and Gyftopoulos +[9], and developed by Beretta [10, 11] exists. This formal- +ism originally sought to establish the second law of ther- +modynamics as a foundational principle at par with other +conservation laws, embedded in a nonlinear law of evo- +lution for mixed states, thus conceptualizing stability of +canonical states and spontaneous decoherence. Later, the +steepest entropy ascent (SEA) formalism was developed +as a powerful modeling tool, applied to various quantum +systems [12, 13], and even elevated to the status of fourth +law of thermodynamics [14, 15] because of its equivalence +with most of the recent far-nonequilibrium modeling for- +malisms. Nonetheless, while discussing possible nonlin- +ear theories of QM in the context of no-signaling, SEA +formalism is hardly ever mentioned in the literature cited +in the preceding paragraph. The authors in Refs. [11, 12] +claim that the SEA formalism abides by the no-signaling +criteria as discussed by Gisin and others. +Yet, an ex- +plicit focus on no-signaling is lacking in their works. As +a result, a lacuna remains to be breached to answer Wein- +berg’s original question. This Letter provides definitive +proof that the SEA nonlinear extension of QM respects +the no-signaling principle. +Philosophically, SEA evolution was designed as part of +a theory whereby spontaneous decoherence could be con- +ceived as a fundamental dynamical feature, in contrast +to the coarse-graining approach. +Entropy, the second +law, and irreversibility could acquire a more fundamen- +tal stature, by emerging as deterministic consequences +of the law of evolution, without contradicting standard +QM [16]. Nonlinearity was known to be essential for this +purpose [17]. While the prevalent notion is that the sec- +ond law is statistical, the pioneers of SEA believed that +the many knots of thermodynamics [17, 18] could be un- +tied by elevating it to a more fundamental stature [9]. +But later, the strength and generality of its mathemat- +ical formalism made SEA a suitable tool for thermody- +namically consistent nonequilibrium modeling even be- +yond and outside of its original context, including the +current quest for fundamental or phenomenological non- +linear extensions of QM for quantum computing applica- +tions, and attempts to assign ontological status to mixed +density operators, albeit motivated differently from Refs. +[1, 9, 16–19]. The ontological hypothesis amounts to as- +suming, accepting, and interpreting the elevation of the +mixed density operators to the same status of ontic phys- +ical states that traditional von Neumann QM reserves to +pure states. To see that it is a conceptual prerequisite to +adopt a nonlinear law of evolution for mixed states, let +Wt with ρ′ = Wt(ρ) denote a nonlinear map represent- +ing the evolution ρ → ρ′ after some time t. Consider the +evolution of three states ρ1, ρ2, ρ3 such that ρ1 → ρ′ +1, +ρ2 → ρ′ +2, ρ3 → ρ′ +3, respectively. Further, assume that +ρ3 = wρ1 + (1−w)ρ2 with 0 ≤ w ≤ 1. Due to the nonlin- +earity of Wt, in general ρ′ +3 ̸= wρ′ +1+(1−w)ρ′ +2. This is no is- + +2 +sue only if the ontological hypothesis is accepted, since w +and 1−w do not represent epistemic ignorance anymore. +A sufficient condition [8] for the nonlinear map to be no- +signaling is that it be convex quasilinear, i.e., it always +admits a w′, 0 ≤ w′ ≤ 1, such that ρ′ +3 = w′ρ′ +1+(1−w′)ρ′ +2. +In this Letter, we introduce a much broader class of non- +convex-quasilinear, nonlinear maps that are no-signaling +The SEA map, despite its logarithmic nonlinearity and +structured construction, belongs to this class and has +general thermodynamic compatibility features. +The no-signaling condition, as noted in [7], is usually +imposed by asking that in the absence of mutual interac- +tions between subsystems A and B, the evolution of the +local observables of A should only depend on its own re- +duced state. The SEA formalism, however, demonstrates +that we can take a less restrictive view [11]: we only re- +quire that, if A and B are non-interacting, the law of +evolution must not allow that a local unitary operation +within B could affect the time evolution of local (reduced, +marginal) state of A. Thus, the condition ρA = ρ′ +A, +such as for the two different states ρ ̸= ρA ⊗ ρB and +ρ′ = ρA ⊗ ρB, does not require that dρA/dt = dρ′ +A/dt, +because local memory of past interactions, i.e., existing +entanglement and/or correlations, may well influence the +local evolutions without violating no-signaling. This in- +corporates the idea that (1) by studying the local evo- +lutions we can disclose the existence of correlations, but +only of the type that can be classically communicated +between the subsystems, and (2) in the absence of inter- +actions the nonlinear dynamics may produce the fading +away of correlations (spontaneous decoherence) but can- +not create new correlations. +In linear QM, the system’s composition is specified by +declaring: (1) the Hilbert space structure as direct prod- +uct H = �M +J=1 HJ of the subspaces of the M compo- +nent subsystems, and (2) the overall Hamiltonian oper- +ator H = �M +J=1 HJ ⊗ IJ + V where HJ (on HJ) is the +local Hamiltonian of the J-th subsystem, IJ the identity +on the direct product HJ = � +K̸=J HK of all the other +subspaces, and V (on H) is the interaction Hamiltonian. +The linear law of evolution, ˙ρ = −i[H, ρ]/ℏ, has a univer- +sal structure and entails the local evolutions through par- +tial tracing, ˙ρJ = −i[HJ, ρJ]/ℏ − i TrJ([V, ρ])/ℏ. Thus, +we recover the universal law ˙ρJ = −i[HJ, ρJ]/ℏ for the +local density operator ρJ = TrJ(ρ) if subsystem J does +not interact with the others (i.e., if V = IJ ⊗ VJ). +Instead, a fully nonlinear QM cannot have a universal +structure, because the subdivision into subsystems must +explicitly concur to the structure of the dynamical law +(see [11] for more on this). A different subdivision re- +quires a different equation of motion. This high price for +abandoning linearity is clearly reflected in the nontrivial +structure of the SEA law of evolution. But this renders it +compatible with the compelling constraint that correla- +tions should not build up and signaling between subsys- +tems should not occur other than per effect of the inter- +action Hamiltonian V through the standard Schr¨odinger +term −i[H, ρ]/ℏ in the evolution law. +Seldom used in composite quantum dynamics analysis, +but crucial in our opinion, are the physical observables +first introduced in [10] and called ‘local perception’ op- +erators (on HJ). Together with their ‘deviation from the +local mean value’ operators and covariance functionals, +they are defined as follows +(X)J +ρ = TrJ[(IJ ⊗ ρJ)X] , +(1) +∆(X)J +ρ = (X)J +ρ − IJ Tr +� +ρJ(X)J +ρ +� +, +(2) +(X, Y )J +ρ = 1 +2 Tr +� +ρJ +� +∆(X)J +ρ, ∆(Y )J +ρ +�� +, +(3) +where ρJ = TrJ(ρ). For a bipartite system AB, the local +perception operators (X)A +ρ (on HA) and (X)B +ρ (on HB) +are the unique operators that for a given X on HAB +satisfy for all states ρ the identity +Tr +� +ρA(X)A +ρ +� += Tr[(ρA ⊗ ρB)X] = Tr +� +ρB(X)B +ρ +� +, +(4) +which shows that they represent all that A and B can +say about the overall observable X by classically sharing +their local states. Operator (X)A +ρ can be viewed as the +projection onto HA of the operator X weighted by the +local state ρB of subsystem B. It is a local observable +for subsystem A which, however, depends on the overall +state ρ and overall observable X. Its local mean value +TrA[ρA(X)A +ρ ] differs from the mean value Tr(ρX) for the +overall system AB, except when A and B are uncorrelated +(ρ = ρA ⊗ ρB). It was dubbed ‘local perception’ because +even if B performs a local tomography and sends the +measured ρB to A by classical communication, the most +that A can measure locally about the overall observable +X is (X)A +ρ . +The overall energy and entropy of the composite sys- +tem are locally perceived within subsystem J through +the operators (H)J +ρ and (S(ρ))J +ρ defined on HJ by Eq. +(1), respectively with X = H, the overall Hamiltonian, +and X = S(ρ) = −kBBln(ρ), that we call the over- +all entropy operator, where Bln(x) denotes the discon- +tinuous function Bln(x) = ln(x) for 0 < x ≤ 1 and +Bln(0) = 0. +Note that the ‘locally perceived over- +all entropy’ operator (S(ρ))J +ρ is different from the ‘lo- +cal entropy’ operator S(ρJ) = −kBBJln(ρJ). +Their +mean values Tr +� +ρJ(S(ρ))J +ρ +� += −kB Tr[(ρJ ⊗ ρJ)Bln(ρ)] +and Tr[ρJS(ρJ)] = −kB Tr[ρJ ln(ρJ)] are different. Only +when ρ = ρJ ⊗ ρJ they are related by Tr +� +ρJ(S(ρ))J +ρ +� += +Tr[ρJS(ρJ)] + Tr[ρJS(ρJ)] = −kB Tr[ρ ln(ρ)]. Likewise, +the ‘locally perceived overall Hamiltonian’ operator (H)J +ρ +is different from the ‘local Hamiltonian’ operator HJ. +Their mean values Tr +� +ρJ(H)J +ρ +� += Tr[(ρJ ⊗ ρJ)H] and +Tr(ρJHJ) are different, and only when V = IJ ⊗ VJ they + +3 +are related by Tr +� +ρJ(H)J +ρ +� += Tr(ρJHJ) + Tr(ρJHJ) = +Tr(ρH). However, it is noteworthy that when the overall +observable X is ‘separable for subsystem J’, in the sense +that X = XJ ⊗IJ +IJ ⊗XJ then, even if ρ ̸= ρJ ⊗ρJ, the +deviations and covariances reduce to their local versions, +∆(X)J +ρ = ∆XJ = XJ − IJ Tr[ρJXJ] , +(5) +(X, Y )J +ρ = Tr[ρJ{∆XJ, ∆YJ}]/2 . +(6) +Now, to formalize the no-signaling definition following +[11] as discussed above, we impose that if A and B are +non-interacting, a local unitary operation on B should +not affect the evolution of A. So, assume that with AB +in the state, ρ a local operation on B changes the state +to +ρ′ = (IA ⊗ UB) ρ (IA ⊗ U † +B) , +(7) +where UB is an arbitrary unitary operator (U † +BUB = IB). +Using the properties of the partial trace, in particular, +TrB[(IA ⊗ XB)ZAB] = TrB[ZAB(IA ⊗ XB)] , +TrA[(IA ⊗ XB)ZAB(IA ⊗ YB)] = XB TrA(ZAB)YB , +we obtain the identities +ρB = TrA[(IA ⊗ U † +B) ρ′ (IA ⊗ UB)] = U † +Bρ′ +BUB, +(8) +ρ′ +A = TrB[(IA ⊗ UB) ρ (IA ⊗ U † +B)] = TrB[(IA ⊗ U † +BUB) ρ] += TrB[(IA ⊗ IB) ρ] = ρA, +(9) +which confirms that a local operation on B does not af- +fect the local state ρA of A, hence the usual idea [7] that +for no-signaling it is sufficient that the dynamical model +implies evolutions of local observables that depend only +on ρA. But it is seldom noted that this is not a necessary +condition. In fact, we prove next that not only the local +reduced state ρA but also the local perception operators +(F(ρ))A of any well-defined nonlinear function F(ρ) of +the overall state (such as the function S(ρ) defined above +for entropy) are not affected by local operations on B ac- +cording to Eq. (7). And since the SEA formalism is based +on such local perception operators, this is an important +lemma in the proof that SEA is no-signaling. +So, let us apply Eq. (7) to a function of F(ρ) as locally +perceived by A represented, according to definition Eq. +(1), by its partial trace weighted with respect to ρB, +(F(ρ))A = TrB[(IA ⊗ ρB)F(ρ)]. +(10) +A function of ρ is defined from its eigenvalue decompo- +sition by F(ρ) = V F(D)V † = � +j F(λj) |λj⟩⟨λj|, where +ρ = V DV †, D = � +j λj |j⟩⟨j|, and V = � +j |λj⟩⟨j|. Since +unitary transformations do not alter the eigenvalues, +F(ρ′) = V ′F(D)V ′† where V ′ = (IA ⊗ UB)V , +(11) +and therefore, using Eq. (8) in the last step, we obtain +(F(ρ′))A = TrB[(IA ⊗ ρ′ +B)F(ρ′)] += TrB[(IA ⊗ ρ′ +B) (IA ⊗ UB)V F(D)V †(IA ⊗ U † +B)] += TrB[(IA ⊗ U † +Bρ′ +BUB) V F(D)V †] += TrB[(IA ⊗ ρB) F(ρ)] = (F(ρ))A . +(12) +This confirms that local operations on B do not affect +the local perception operators of A and, therefore, their +proper use in nonlinear QM does not cause signaling is- +sues. +We are now ready to introduce the last but not least +essential ingredient of a general composite-system non- +linear QM, namely, the system’s structure-dependent ex- +pressions of the separate contribution of each subsystem +to the dissipative term of the equation of motion for the +overall state ρ. As discussed above (and clearly recog- +nized in the early SEA literature [10, 11]), the composite- +system nonlinear evolution should reflect explicitly the +internal structure of the system, essentially by declaring +which subsystems are to be prevented from nonphysical +effects such as signaling, exchange of energy, or build- +up of correlations between non-interacting subsystems. +In terms of the notation introduced above, the structure +proposed in [10, 11] for the dissipative term of the dy- +namics to be added to the usual Hamiltonian term is as +follows +dρ +dt = − i +ℏ[H , ρ] − +M +� +J=1 +{DJ +ρ , ρJ} ⊗ ρJ , +(13) +where the ‘local dissipation operators’ DJ +ρ (on HJ) may +be nonlinear functions of the local observables of J, the +reduced state ρJ, and the local perception operators of +overall observables. For the dissipative term to preserve +Tr(ρ), operators +� +DJ +ρ , ρJ +� +must be traceless. +To pre- +serve Tr(ρH) [and possibly other conserved properties +Tr(ρCk)], operators +� +DJ +ρ , ρJ +� +(H)J +ρ [and +� +DJ +ρ , ρJ +� +(Ck)J +ρ ] +must also be traceless. The rate of change of the overall +system entropy s(ρ) = −kB Tr[ρ ln(ρ)] is +ds(ρ) +dt += − +M +� +J=1 +Tr +� +{DJ +ρ , ρJ} (S(ρ))J +ρ +� +, +(14) +and the local nonlinear evolution of subsystem J is ob- +tained by partial tracing over HJ, in general, +dρJ +dt = − i +ℏ[HJ, ρJ] − i +ℏ TrJ([V, ρ] ) − {DJ +ρ , ρJ} , +(15) +where we recall that the second term in the rhs can be +put, for weak interactions and under well-known assump- +tions, in Kossakowski-Lindblad form. +Before introducing the SEA assumption, we emphasize +that the construction obtained so far, Eq. (13), opens + +4 +up and paves the way for a class of no-signaling nonlin- +ear evolution equations that is much broader, through +all the possible compatible choices of the operators DJ +ρ , +than nonlinear laws restricted by the sufficient but not +necessary condition that dρJ/dt be a function of ρJ only. +We can formally state this no-signaling condition using +the following statement, +dρJ +dt = f(ρJ, (Ck)J). +(16) +Finally, to introduce the SEA assumption in the spirit +of the fourth law of thermodynamics [14, 15], one way is +employing a variational principle. We first observe from +Eq. (14) that the rate of entropy change contributed by +subsystem J is directly proportional to the norm of op- +erator DJ +ρ , so there is no maximum entropy production +rate because we can trivially increase it indefinitely by +simple multiplication of DJ +ρ by a positive scalar. But we +can fix that norm, and maximize against the direction +in operator space, to identify, for each given state ρ, the +operators DJ +ρ that point in the direction of steepest en- +tropy ascent. To this end, to recover the original SEA +formulation [10] let us maximize Eq. (14) subject to the +conservation constraints Tr +� +{DJ +ρ , ρJ} (Ck)J +ρ +� += 0 where +C1 = I, C2 = H, and Ck are other conserved prop- +erties (if any), together with the fixed weighted norm +constraints Tr +� +ρJ(DJ +ρ )2� += const (for more general SEA +formulations in terms of a different metric as necessary to +incorporate Onsager reciprocity see [14, 15]). Introduc- +ing Lagrange multipliers βJ +k and τJ for the conservation +and norm constraints, respectively, and imposing vanish- +ing variational derivatives with respect to operators DJ +ρ +at fixed ρ and ρJ’s (derivation details in [11, 14]) yields +2τJDJ +ρ = (Bln(ρ))J +ρ + � +ℓβJ +ℓ (Cℓ)J +ρ . +(17) +where the multipliers βJ +ℓ must solve the system of equa- +tions obtained by substituting these maximizing expres- +sions of the DJ +ρ ’s into the conservation constraints, +� +ℓ +βJ +ℓ Tr +� +ρJ +� +(Cℓ)J +ρ , (Ck)J +ρ +�� += − Tr +� +ρJ +� +(Bln(ρ))J +ρ , (Ck)J +ρ +�� +. +(18) +When C1 = I and C2 = H determine the conserved +properties and Eqs. (18) are linearly independent, using +Cramers’ rule, properties of determinants, and definitions +(2) and (3), the SEA dissipators can be cast as +DJ +ρ = +1 +4τJ +������ +∆(Bln(ρ))J +ρ +∆(H)J +ρ +(H, Bln(ρ))J +ρ (H, H)J +ρ +������ +(H, H)J +ρ +. +(19) +The rate of entropy production may be expressed as +ds(ρ) +dt += +M +� +J=1 +1 +2τJ +������ +(Bln(ρ), Bln(ρ))J +ρ (H, Bln(ρ))J +ρ +(H, Bln(ρ))J +ρ +(H, H)J +ρ +������ +(H, H)J +ρ +, +(20) +showing clearly that it is nonnegative since the numera- +tors in the summation are Gram determinants. +Regarding no-signaling, we note that: (1) if subsys- +tem J is noninteracting, H = HJ ⊗ IJ + IJ ⊗ HJ, +then ∆(H)J +ρ += +HJ − IJ Tr(ρJHJ) and (H, H)J +ρ += +Tr +� +ρJ(∆HJ)2� +depend only on the local HJ and ρJ; and +(2) if J is uncorrelated, Bln(ρ) = Bln(ρJ) ⊗ IJ + IJ ⊗ +Bln(ρJ), then ∆(Bln(ρ))J +ρ = Bln(ρJ) − IJ Tr(ρJ ln(ρJ)) +and (Bln(ρ), Bln(ρ))J +ρ = Tr +� +ρJ(ln(ρJ))2� +depend only on +the local ρJ. Therefore, it is only when J is both nonin- +teracting and uncorrelated that its local dissipation op- +erator DJ +ρ depends only on the local HJ and ρJ, and +the local equation of motion Eq. (15) reduces exactly to +the non-composite system version of SEA evolution [10]. +Instead, if J is either interacting or correlated, DJ +ρ and, +therefore, the local nonlinear SEA evolution according to +Eq. (15), is determined not only by the local HJ and ρJ, +but also by the local perceptions of the overall interaction +Hamiltonian and/or the overall entropy operator Bln(ρ), +nonetheless without violating the no-signaling condition. +In extremal cases, it is known [10, 11, 20] that even +if the subsystems are entangled and therefore the local +states ρJ are mixed, operators DJ +ρ vanish and Eqs. (13) +and (15) reduce to the standard Schr¨odinger equation. +E.g., if the overall system is in a pure state, Bln(ρ) = 0, +standard unitary evolutions of pure states emerge as limit +cycles of the nonlinear SEA dynamics. +Consider the example of a two-qubit composite AB. +The mixed and correlated states +ρ = 1 +4 +� +I4+ +� +j={x,y,z} +(aj σj⊗I2+bj I2⊗σj+cj σj⊗σj) +� +, (21) +are Bell diagonal states if aj = bj = 0 for all j’s (and +Werner states if in addition cj = 4w/3−1 for all j’s) with +eigenvalues 4λ1 = 1 − cx − cy − cz, 4λ2 = 1 − cx + cy + cz, +4λ3 = 1 + cx − cy + cz, 4λ4 = 1 + cx + cy − cz. Somewhat +surprisingly, Bell diagonal states are nondissipative limit +cycles within nonlinear SEA dynamics, under any Hamil- +tonian. Indeed, we find (Bln(ρBell))J +ρ = I2 +� +k Bln(λk)/2, +so that ∆(Bln(ρ))J +ρ = 0 and DJ +ρ = 0, for both J = A, B. +But most neighboring and other states in this class are +dissipative. +For a simple example of correlated but separable mixed +states, assume ax = a, bx = b, and ay = az = by = bz = +cx = cy = cz = 0, so that ρA ⊗ ρB − ρ = (ab/4)σx ⊗ σx +and the eigenvalues are 4λ1 = 1 − a − b, 4λ2 = 1 − a + b, +4λ3 = 1+a−b, 4λ4 = 1+a+b. If the two noninteracting + +5 +qubits A and B have local Hamiltonians HA = HB = σz, +we find +{DA +ρ , ρA} = (1 − a2) +16 +(bfa,b − ga,b) σx, +(22) +{DB +ρ , ρB} = (1 − b2) +16 +(afa,b − ha,b) σx, +(23) +where fa,b = Bln(λ1) − Bln(λ2) − Bln(λ3) + Bln(λ4), +ga,b = Bln(λ1) + Bln(λ2) − Bln(λ3) − Bln(λ4), ha,b = +Bln(λ1) − Bln(λ2) + Bln(λ3) − Bln(λ4) so that the non- +linear evolution is clearly nontrivial. But it preserves the +zero mean energies of both qubits, and while the overall +entropy increases and mutual information partially fades +away, it drives the overall state towards a nondissipative +correlated state with maximally mixed marginals. We +proved above that signaling is impossible, even though +DA +ρ depends not only on a but also on b, and DB +ρ on a +which agrees with our no-signaling condition in Eq. (16). +For a slightly more elaborate example that includes +entangled mixed states, assume ax = az = a/ +√ +2, bx = +bz = b/ +√ +2, and cx = cy = cz = 2(a − b)/3, so that +the eigenvalues are 4λ1 = 1 + a − b, 12λ2 = 3 − a − 5b, +12λ3 = 3 + 5a + b, 12λ4 = 3 − 7a + 7b, and those of the +partial tranpose 12λP T +1 += 3 + a− b, 12λP T +2 += 3 − 5a+ 5b, +12λP T +3 += 3 + 2a − 2b + +√ +d, 12λP T +4 += 3 + 2a − 2b − +√ +d +with d = 25a2 − 14ab + 25b2. For a = −b these states +are separable for −3/14 ≤ b ≤ 1/4 and entangled for +1/4 < b ≤ 1/2. If the two noninteracting qubits A and B +have local Hamiltonians HA = HB = σz, we find +{DA +ρ , ρA} = +√ +2(1 − a2) +80(2 − a2) (fa,b − 5bha,b) σx, +(24) +{DB +ρ , ρB} = − +√ +2(1 − b2) +80(2 − b2) (ga,b + 5aha,b) σx, +(25) +where here fa,b = 3Bln(λ1) − 5Bln(λ2) + 5Bln(λ3) − +3Bln(λ4), ga,b = 3Bln(λ1) + 5Bln(λ2) − 5Bln(λ3) − +3Bln(λ4), ha,b = Bln(λ1)−Bln(λ2)−Bln(λ3)+Bln(λ4) so +that again the nonlinear evolution is clearly nontrivial in +the sense that the local nonlinear evolution of A (B) does +not depend only on ρA (ρB), despite being no-signaling. +To summarize, in this Letter we prove that the SEA +formalism provides a valid non-linear extension of QM. +To show this, we explore the definition of no-signaling +for composite systems and provide generalized necessary +criteria in terms of locally perceived operators, less re- +strictive than the traditional criterion in terms of local +density operators. Furthermore, we build on that defini- +tion and show how, by construction, SEA is no-signaling. +For non-interacting subsystems, the traditional criterion +is met for uncorrelated states, but we provide nontrivial +examples of correlated states for which it is not met. +RKR is grateful to the INSPIRE Fellowship program +by the Department of Science and Technology, Govt. of +India for funding his Ph.D., to Prof. Alok Pan of the In- +dian Institute of Technology Hyderabad for many useful +discussions, and to the Wolfram Publication Watch Team +for providing full access to online Mathematica [21]. +∗ rohitkray@iitkgp.ac.in +† gianpaolo.beretta@unibs.it +[1] S. Weinberg, Annals of Physics 194, 336 (1989). +[2] N. Gisin, Physics Letters A 143, 1 (1990). +[3] J. Polchinski, Phys. Rev. Lett. 66, 397 (1991). +[4] J. L. Park, Found Phys 1, 23 (1970). +[5] W. +K. +Wootters +and +W. +H. +Zurek, +Nature 299, 802 (1982); +D. +Dieks, +Physics Letters A 92, 271 (1982). +[6] N. +Gisin +and +M. +Rigo, +J. Phys. A: Math. Gen. 28, 7375 (1995). +[7] M. Ferrero, D. Salgado, and J. L. S´anchez-G´omez, +Phys. Rev. A 70, 014101 (2004). +[8] J. +Rembieli´nski +and +P. +Caban, +Phys. Rev. Research 2, 012027 (2020); +Quantum 5, 420 (2021), arXiv:2003.09170 [quant-ph]. +[9] G. +N. +Hatsopoulos +and +E. +P. +Gyftopou- +los, +Foundations of Physics 6, 15 (1976); +Foundations of Physics 6, 127 (1976); +Foundations of Physics 6, 439 (1976); +Foundations of Physics 6, 561 (1976). +[10] G. P. Beretta, E. P. Gyftopoulos, J. L. Park, and +G. N. Hatsopoulos, Nuovo Cimento B 82, 169 (1984); +G. +P. +Beretta, +E. +P. +Gyftopoulos, +and +J. +L. +Park, +Nuovo Cimento B 87, 77 (1985); +J. +Maddox, +Nature 316, 11 (1985). +[11] G. P. Beretta, Modern Physics Letters A 20, 977 (2005); +Reports on Mathematical Physics 64, 139 (2009); +Journal of Physics: Conference Series 237, 012004 (2010); +Entropy 21, 679 (2019). +[12] S. +Cano-Andrade, +G. +P. +Beretta, +and +M. +R. +von +Spakovsky, +Phys. Rev. A 91, 013848 (2015); +M. +R. +Von +Spakovsky +and +J. +Gemmer, +Entropy 16, 3434 (2014). +[13] R. K. Ray, Phys. Rev. E 106, 024115 (2022). +[14] G. P. Beretta, Phys. Rev. E 90, 042113 (2014). +[15] G. P. Beretta, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, 20190168 (2020). +[16] G. P. Beretta, Intrinsic entropy and intrinsic irre- +versibility for a single isolated constituent of matter: +Broader kinematics and generalized nonlinear dynam- +ics, +in +Frontiers of Nonequilibrium Statistical Physics, +edited by G. T. Moore and M. O. Scully (Springer US, +Boston, MA, 1986) pp. 205–212. +[17] R. +F. +Simmons +and +J. +L. +Park, +Foundations of Physics 11, 297 (1981); +J. +L. +Park +and +R. +F. +Simmons, +The +knots +of +quantum +thermodynamics, +in +Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology: Essays in Honor of Wolfgang Yourgrau, +edited by A. van der Merwe (Springer US, Boston, MA, +1983) pp. 289–308. +[18] G. +N. +Hatsopoulos +and +G. +P. +Beretta, +AIP Conference Proceedings 1033, 34 (2008). +[19] G. P. Beretta, Modern Physics Letters A 21, 2799 (2006). +[20] G. P. Beretta, Phys. Rev. E 73, 026113 (2006). +[21] Wolfram Research Inc., Mathematica Online (2022). + diff --git a/ttFJT4oBgHgl3EQfdiwx/content/tmp_files/load_file.txt b/ttFJT4oBgHgl3EQfdiwx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7f3f12bb7fdfd559b4ca4e22f52aaff8b233b5f --- /dev/null +++ b/ttFJT4oBgHgl3EQfdiwx/content/tmp_files/load_file.txt @@ -0,0 +1,302 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf,len=301 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='11548v1 [quant-ph] 27 Jan 2023 No-signaling in Nonlinear Extensions of Quantum Mechanics Rohit Kishan Ray∗ Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Indian Institute of Technology Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Kharagpur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' West Bengal - 721302,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' India Gian Paolo Beretta† Department of Mechanical and Industrial Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' University of Brescia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' 25123 Brescia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Italy (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' 2023) Devising a nonlinear extension of quantum mechanics is nontrivial because unphysical features such as supraluminal communication (signaling) are to be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In this Letter, we show that the steepest entropy ascent formalism is a viable no-signaling extension belonging to a broader class of no-signaling nonlinear evolution equations for which the local evolution of a subsystem is not necessarily bound to depend only on its reduced state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Quantum mechanics (QM) in its most common form (Schr¨odinger–von Neumann formalism) is linear in state space, where linear operators operate upon state vectors, and the time evolution is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In 1989, when the late Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Weinberg asked to test the linearity of quantum mechanics as we know it [1], a new quest began for the physicists working in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Thence onward, distin- guished researchers such as Gisin [2] and Polchinski [3] showed that introducing non-linearity via operators pro- duces signaling (faster than light communication), lead- ing to the violation of causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In this context as a consequence of using linear operators in QM, we must take note of the fact that no-cloning, first introduced by Park [4] and later explicitly shown by Wootters, Zurek, and Dieks [5], is sufficient for no-signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Nonlinear- ity, when introduced via stochastic QM through Lind- blad operator formalism [6] for open quantum systems, has been shown to respect no-signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Ensuing work by Ferrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' [7] has shown that nonlinearity in QM can be accommodated without compromising no-signaling if we consider the time evolution to be nonlinear albeit maintaining linearity in the state space and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In a more recent work [8], it has been shown that convex quasilinear maps can contribute to the nonlinear dynam- ics of QM without invoking signaling, and this formalism retains much of QM as it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Thusly, Rembieli´nski and Caban [8] have found a minimal permissible deviation from the linear structure of QM dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For about four decades, a nonlinear extension of quan- tum mechanics initiated by Hatsopoulos and Gyftopoulos [9], and developed by Beretta [10, 11] exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' This formal- ism originally sought to establish the second law of ther- modynamics as a foundational principle at par with other conservation laws, embedded in a nonlinear law of evo- lution for mixed states, thus conceptualizing stability of canonical states and spontaneous decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Later, the steepest entropy ascent (SEA) formalism was developed as a powerful modeling tool, applied to various quantum systems [12, 13], and even elevated to the status of fourth law of thermodynamics [14, 15] because of its equivalence with most of the recent far-nonequilibrium modeling for- malisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Nonetheless, while discussing possible nonlin- ear theories of QM in the context of no-signaling, SEA formalism is hardly ever mentioned in the literature cited in the preceding paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The authors in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' [11, 12] claim that the SEA formalism abides by the no-signaling criteria as discussed by Gisin and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Yet, an ex- plicit focus on no-signaling is lacking in their works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' As a result, a lacuna remains to be breached to answer Wein- berg’s original question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' This Letter provides definitive proof that the SEA nonlinear extension of QM respects the no-signaling principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Philosophically, SEA evolution was designed as part of a theory whereby spontaneous decoherence could be con- ceived as a fundamental dynamical feature, in contrast to the coarse-graining approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Entropy, the second law, and irreversibility could acquire a more fundamen- tal stature, by emerging as deterministic consequences of the law of evolution, without contradicting standard QM [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Nonlinearity was known to be essential for this purpose [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' While the prevalent notion is that the sec- ond law is statistical, the pioneers of SEA believed that the many knots of thermodynamics [17, 18] could be un- tied by elevating it to a more fundamental stature [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' But later, the strength and generality of its mathemat- ical formalism made SEA a suitable tool for thermody- namically consistent nonequilibrium modeling even be- yond and outside of its original context, including the current quest for fundamental or phenomenological non- linear extensions of QM for quantum computing applica- tions, and attempts to assign ontological status to mixed density operators, albeit motivated differently from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' [1, 9, 16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The ontological hypothesis amounts to as- suming, accepting, and interpreting the elevation of the mixed density operators to the same status of ontic phys- ical states that traditional von Neumann QM reserves to pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' To see that it is a conceptual prerequisite to adopt a nonlinear law of evolution for mixed states, let Wt with ρ′ = Wt(ρ) denote a nonlinear map represent- ing the evolution ρ → ρ′ after some time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Consider the evolution of three states ρ1, ρ2, ρ3 such that ρ1 → ρ′ 1, ρ2 → ρ′ 2, ρ3 → ρ′ 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Further, assume that ρ3 = wρ1 + (1−w)ρ2 with 0 ≤ w ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Due to the nonlin- earity of Wt, in general ρ′ 3 ̸= wρ′ 1+(1−w)ρ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' This is no is- 2 sue only if the ontological hypothesis is accepted, since w and 1−w do not represent epistemic ignorance anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' A sufficient condition [8] for the nonlinear map to be no- signaling is that it be convex quasilinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=', it always admits a w′, 0 ≤ w′ ≤ 1, such that ρ′ 3 = w′ρ′ 1+(1−w′)ρ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In this Letter, we introduce a much broader class of non- convex-quasilinear, nonlinear maps that are no-signaling The SEA map, despite its logarithmic nonlinearity and structured construction, belongs to this class and has general thermodynamic compatibility features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The no-signaling condition, as noted in [7], is usually imposed by asking that in the absence of mutual interac- tions between subsystems A and B, the evolution of the local observables of A should only depend on its own re- duced state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The SEA formalism, however, demonstrates that we can take a less restrictive view [11]: we only re- quire that, if A and B are non-interacting, the law of evolution must not allow that a local unitary operation within B could affect the time evolution of local (reduced, marginal) state of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Thus, the condition ρA = ρ′ A, such as for the two different states ρ ̸= ρA ⊗ ρB and ρ′ = ρA ⊗ ρB, does not require that dρA/dt = dρ′ A/dt, because local memory of past interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=', existing entanglement and/or correlations, may well influence the local evolutions without violating no-signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' This in- corporates the idea that (1) by studying the local evo- lutions we can disclose the existence of correlations, but only of the type that can be classically communicated between the subsystems, and (2) in the absence of inter- actions the nonlinear dynamics may produce the fading away of correlations (spontaneous decoherence) but can- not create new correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In linear QM, the system’s composition is specified by declaring: (1) the Hilbert space structure as direct prod- uct H = �M J=1 HJ of the subspaces of the M compo- nent subsystems, and (2) the overall Hamiltonian oper- ator H = �M J=1 HJ ⊗ IJ + V where HJ (on HJ) is the local Hamiltonian of the J-th subsystem, IJ the identity on the direct product HJ = � K̸=J HK of all the other subspaces, and V (on H) is the interaction Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The linear law of evolution, ˙ρ = −i[H, ρ]/ℏ, has a univer- sal structure and entails the local evolutions through par- tial tracing, ˙ρJ = −i[HJ, ρJ]/ℏ − i TrJ([V, ρ])/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Thus, we recover the universal law ˙ρJ = −i[HJ, ρJ]/ℏ for the local density operator ρJ = TrJ(ρ) if subsystem J does not interact with the others (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=', if V = IJ ⊗ VJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Instead, a fully nonlinear QM cannot have a universal structure, because the subdivision into subsystems must explicitly concur to the structure of the dynamical law (see [11] for more on this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' A different subdivision re- quires a different equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' This high price for abandoning linearity is clearly reflected in the nontrivial structure of the SEA law of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' But this renders it compatible with the compelling constraint that correla- tions should not build up and signaling between subsys- tems should not occur other than per effect of the inter- action Hamiltonian V through the standard Schr¨odinger term −i[H, ρ]/ℏ in the evolution law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Seldom used in composite quantum dynamics analysis, but crucial in our opinion, are the physical observables first introduced in [10] and called ‘local perception’ op- erators (on HJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Together with their ‘deviation from the local mean value’ operators and covariance functionals, they are defined as follows (X)J ρ = TrJ[(IJ ⊗ ρJ)X] , (1) ∆(X)J ρ = (X)J ρ − IJ Tr � ρJ(X)J ρ � , (2) (X, Y )J ρ = 1 2 Tr � ρJ � ∆(X)J ρ, ∆(Y )J ρ �� , (3) where ρJ = TrJ(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For a bipartite system AB, the local perception operators (X)A ρ (on HA) and (X)B ρ (on HB) are the unique operators that for a given X on HAB satisfy for all states ρ the identity Tr � ρA(X)A ρ � = Tr[(ρA ⊗ ρB)X] = Tr � ρB(X)B ρ � , (4) which shows that they represent all that A and B can say about the overall observable X by classically sharing their local states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Operator (X)A ρ can be viewed as the projection onto HA of the operator X weighted by the local state ρB of subsystem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' It is a local observable for subsystem A which, however, depends on the overall state ρ and overall observable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Its local mean value TrA[ρA(X)A ρ ] differs from the mean value Tr(ρX) for the overall system AB, except when A and B are uncorrelated (ρ = ρA ⊗ ρB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' It was dubbed ‘local perception’ because even if B performs a local tomography and sends the measured ρB to A by classical communication, the most that A can measure locally about the overall observable X is (X)A ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The overall energy and entropy of the composite sys- tem are locally perceived within subsystem J through the operators (H)J ρ and (S(ρ))J ρ defined on HJ by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (1), respectively with X = H, the overall Hamiltonian, and X = S(ρ) = −kBBln(ρ), that we call the over- all entropy operator, where Bln(x) denotes the discon- tinuous function Bln(x) = ln(x) for 0 < x ≤ 1 and Bln(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Note that the ‘locally perceived over- all entropy’ operator (S(ρ))J ρ is different from the ‘lo- cal entropy’ operator S(ρJ) = −kBBJln(ρJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Their mean values Tr � ρJ(S(ρ))J ρ � = −kB Tr[(ρJ ⊗ ρJ)Bln(ρ)] and Tr[ρJS(ρJ)] = −kB Tr[ρJ ln(ρJ)] are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Only when ρ = ρJ ⊗ ρJ they are related by Tr � ρJ(S(ρ))J ρ � = Tr[ρJS(ρJ)] + Tr[ρJS(ρJ)] = −kB Tr[ρ ln(ρ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Likewise, the ‘locally perceived overall Hamiltonian’ operator (H)J ρ is different from the ‘local Hamiltonian’ operator HJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Their mean values Tr � ρJ(H)J ρ � = Tr[(ρJ ⊗ ρJ)H] and Tr(ρJHJ) are different, and only when V = IJ ⊗ VJ they 3 are related by Tr � ρJ(H)J ρ � = Tr(ρJHJ) + Tr(ρJHJ) = Tr(ρH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' However, it is noteworthy that when the overall observable X is ‘separable for subsystem J’, in the sense that X = XJ ⊗IJ +IJ ⊗XJ then, even if ρ ̸= ρJ ⊗ρJ, the deviations and covariances reduce to their local versions, ∆(X)J ρ = ∆XJ = XJ − IJ Tr[ρJXJ] , (5) (X, Y )J ρ = Tr[ρJ{∆XJ, ∆YJ}]/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (6) Now, to formalize the no-signaling definition following [11] as discussed above, we impose that if A and B are non-interacting, a local unitary operation on B should not affect the evolution of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' So, assume that with AB in the state, ρ a local operation on B changes the state to ρ′ = (IA ⊗ UB) ρ (IA ⊗ U † B) , (7) where UB is an arbitrary unitary operator (U † BUB = IB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Using the properties of the partial trace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' TrB[(IA ⊗ XB)ZAB] = TrB[ZAB(IA ⊗ XB)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' TrA[(IA ⊗ XB)ZAB(IA ⊗ YB)] = XB TrA(ZAB)YB ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' we obtain the identities ρB = TrA[(IA ⊗ U † B) ρ′ (IA ⊗ UB)] = U † Bρ′ BUB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (8) ρ′ A = TrB[(IA ⊗ UB) ρ (IA ⊗ U † B)] = TrB[(IA ⊗ U † BUB) ρ] = TrB[(IA ⊗ IB) ρ] = ρA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (9) which confirms that a local operation on B does not af- fect the local state ρA of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' hence the usual idea [7] that for no-signaling it is sufficient that the dynamical model implies evolutions of local observables that depend only on ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' But it is seldom noted that this is not a necessary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In fact, we prove next that not only the local reduced state ρA but also the local perception operators (F(ρ))A of any well-defined nonlinear function F(ρ) of the overall state (such as the function S(ρ) defined above for entropy) are not affected by local operations on B ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' And since the SEA formalism is based on such local perception operators, this is an important lemma in the proof that SEA is no-signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' So, let us apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (7) to a function of F(ρ) as locally perceived by A represented, according to definition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (1), by its partial trace weighted with respect to ρB, (F(ρ))A = TrB[(IA ⊗ ρB)F(ρ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (10) A function of ρ is defined from its eigenvalue decompo- sition by F(ρ) = V F(D)V † = � j F(λj) |λj⟩⟨λj|, where ρ = V DV †, D = � j λj |j⟩⟨j|, and V = � j |λj⟩⟨j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Since unitary transformations do not alter the eigenvalues, F(ρ′) = V ′F(D)V ′† where V ′ = (IA ⊗ UB)V , (11) and therefore, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (8) in the last step, we obtain (F(ρ′))A = TrB[(IA ⊗ ρ′ B)F(ρ′)] = TrB[(IA ⊗ ρ′ B) (IA ⊗ UB)V F(D)V †(IA ⊗ U † B)] = TrB[(IA ⊗ U † Bρ′ BUB) V F(D)V †] = TrB[(IA ⊗ ρB) F(ρ)] = (F(ρ))A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (12) This confirms that local operations on B do not affect the local perception operators of A and, therefore, their proper use in nonlinear QM does not cause signaling is- sues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' We are now ready to introduce the last but not least essential ingredient of a general composite-system non- linear QM, namely, the system’s structure-dependent ex- pressions of the separate contribution of each subsystem to the dissipative term of the equation of motion for the overall state ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' As discussed above (and clearly recog- nized in the early SEA literature [10, 11]), the composite- system nonlinear evolution should reflect explicitly the internal structure of the system, essentially by declaring which subsystems are to be prevented from nonphysical effects such as signaling, exchange of energy, or build- up of correlations between non-interacting subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In terms of the notation introduced above, the structure proposed in [10, 11] for the dissipative term of the dy- namics to be added to the usual Hamiltonian term is as follows dρ dt = − i ℏ[H , ρ] − M � J=1 {DJ ρ , ρJ} ⊗ ρJ , (13) where the ‘local dissipation operators’ DJ ρ (on HJ) may be nonlinear functions of the local observables of J, the reduced state ρJ, and the local perception operators of overall observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For the dissipative term to preserve Tr(ρ), operators � DJ ρ , ρJ � must be traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' To pre- serve Tr(ρH) [and possibly other conserved properties Tr(ρCk)], operators � DJ ρ , ρJ � (H)J ρ [and � DJ ρ , ρJ � (Ck)J ρ ] must also be traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The rate of change of the overall system entropy s(ρ) = −kB Tr[ρ ln(ρ)] is ds(ρ) dt = − M � J=1 Tr � {DJ ρ , ρJ} (S(ρ))J ρ � , (14) and the local nonlinear evolution of subsystem J is ob- tained by partial tracing over HJ, in general, dρJ dt = − i ℏ[HJ, ρJ] − i ℏ TrJ([V, ρ] ) − {DJ ρ , ρJ} , (15) where we recall that the second term in the rhs can be put, for weak interactions and under well-known assump- tions, in Kossakowski-Lindblad form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Before introducing the SEA assumption, we emphasize that the construction obtained so far, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (13), opens 4 up and paves the way for a class of no-signaling nonlin- ear evolution equations that is much broader, through all the possible compatible choices of the operators DJ ρ , than nonlinear laws restricted by the sufficient but not necessary condition that dρJ/dt be a function of ρJ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' We can formally state this no-signaling condition using the following statement, dρJ dt = f(ρJ, (Ck)J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (16) Finally, to introduce the SEA assumption in the spirit of the fourth law of thermodynamics [14, 15], one way is employing a variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' We first observe from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (14) that the rate of entropy change contributed by subsystem J is directly proportional to the norm of op- erator DJ ρ , so there is no maximum entropy production rate because we can trivially increase it indefinitely by simple multiplication of DJ ρ by a positive scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' But we can fix that norm, and maximize against the direction in operator space, to identify, for each given state ρ, the operators DJ ρ that point in the direction of steepest en- tropy ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' To this end, to recover the original SEA formulation [10] let us maximize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (14) subject to the conservation constraints Tr � {DJ ρ , ρJ} (Ck)J ρ � = 0 where C1 = I, C2 = H, and Ck are other conserved prop- erties (if any), together with the fixed weighted norm constraints Tr � ρJ(DJ ρ )2� = const (for more general SEA formulations in terms of a different metric as necessary to incorporate Onsager reciprocity see [14, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Introduc- ing Lagrange multipliers βJ k and τJ for the conservation and norm constraints, respectively, and imposing vanish- ing variational derivatives with respect to operators DJ ρ at fixed ρ and ρJ’s (derivation details in [11, 14]) yields 2τJDJ ρ = (Bln(ρ))J ρ + � ℓβJ ℓ (Cℓ)J ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (17) where the multipliers βJ ℓ must solve the system of equa- tions obtained by substituting these maximizing expres- sions of the DJ ρ ’s into the conservation constraints, � ℓ βJ ℓ Tr � ρJ � (Cℓ)J ρ , (Ck)J ρ �� = − Tr � ρJ � (Bln(ρ))J ρ , (Ck)J ρ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (18) When C1 = I and C2 = H determine the conserved properties and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (18) are linearly independent, using Cramers’ rule, properties of determinants, and definitions (2) and (3), the SEA dissipators can be cast as DJ ρ = 1 4τJ ������ ∆(Bln(ρ))J ρ ∆(H)J ρ (H, Bln(ρ))J ρ (H, H)J ρ ������ (H, H)J ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (19) The rate of entropy production may be expressed as ds(ρ) dt = M � J=1 1 2τJ ������ (Bln(ρ), Bln(ρ))J ρ (H, Bln(ρ))J ρ (H, Bln(ρ))J ρ (H, H)J ρ ������ (H, H)J ρ , (20) showing clearly that it is nonnegative since the numera- tors in the summation are Gram determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Regarding no-signaling, we note that: (1) if subsys- tem J is noninteracting, H = HJ ⊗ IJ + IJ ⊗ HJ, then ∆(H)J ρ = HJ − IJ Tr(ρJHJ) and (H, H)J ρ = Tr � ρJ(∆HJ)2� depend only on the local HJ and ρJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' and (2) if J is uncorrelated, Bln(ρ) = Bln(ρJ) ⊗ IJ + IJ ⊗ Bln(ρJ), then ∆(Bln(ρ))J ρ = Bln(ρJ) − IJ Tr(ρJ ln(ρJ)) and (Bln(ρ), Bln(ρ))J ρ = Tr � ρJ(ln(ρJ))2� depend only on the local ρJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Therefore, it is only when J is both nonin- teracting and uncorrelated that its local dissipation op- erator DJ ρ depends only on the local HJ and ρJ, and the local equation of motion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (15) reduces exactly to the non-composite system version of SEA evolution [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Instead, if J is either interacting or correlated, DJ ρ and, therefore, the local nonlinear SEA evolution according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (15), is determined not only by the local HJ and ρJ, but also by the local perceptions of the overall interaction Hamiltonian and/or the overall entropy operator Bln(ρ), nonetheless without violating the no-signaling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' In extremal cases, it is known [10, 11, 20] that even if the subsystems are entangled and therefore the local states ρJ are mixed, operators DJ ρ vanish and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (13) and (15) reduce to the standard Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=', if the overall system is in a pure state, Bln(ρ) = 0, standard unitary evolutions of pure states emerge as limit cycles of the nonlinear SEA dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Consider the example of a two-qubit composite AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' The mixed and correlated states ρ = 1 4 � I4+ � j={x,y,z} (aj σj⊗I2+bj I2⊗σj+cj σj⊗σj) � , (21) are Bell diagonal states if aj = bj = 0 for all j’s (and Werner states if in addition cj = 4w/3−1 for all j’s) with eigenvalues 4λ1 = 1 − cx − cy − cz, 4λ2 = 1 − cx + cy + cz, 4λ3 = 1 + cx − cy + cz, 4λ4 = 1 + cx + cy − cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Somewhat surprisingly, Bell diagonal states are nondissipative limit cycles within nonlinear SEA dynamics, under any Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Indeed, we find (Bln(ρBell))J ρ = I2 � k Bln(λk)/2, so that ∆(Bln(ρ))J ρ = 0 and DJ ρ = 0, for both J = A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' But most neighboring and other states in this class are dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For a simple example of correlated but separable mixed states, assume ax = a, bx = b, and ay = az = by = bz = cx = cy = cz = 0, so that ρA ⊗ ρB − ρ = (ab/4)σx ⊗ σx and the eigenvalues are 4λ1 = 1 − a − b, 4λ2 = 1 − a + b, 4λ3 = 1+a−b, 4λ4 = 1+a+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' If the two noninteracting 5 qubits A and B have local Hamiltonians HA = HB = σz, we find {DA ρ , ρA} = (1 − a2) 16 (bfa,b − ga,b) σx, (22) {DB ρ , ρB} = (1 − b2) 16 (afa,b − ha,b) σx, (23) where fa,b = Bln(λ1) − Bln(λ2) − Bln(λ3) + Bln(λ4), ga,b = Bln(λ1) + Bln(λ2) − Bln(λ3) − Bln(λ4), ha,b = Bln(λ1) − Bln(λ2) + Bln(λ3) − Bln(λ4) so that the non- linear evolution is clearly nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' But it preserves the zero mean energies of both qubits, and while the overall entropy increases and mutual information partially fades away, it drives the overall state towards a nondissipative correlated state with maximally mixed marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' We proved above that signaling is impossible, even though DA ρ depends not only on a but also on b, and DB ρ on a which agrees with our no-signaling condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For a slightly more elaborate example that includes entangled mixed states, assume ax = az = a/ √ 2, bx = bz = b/ √ 2, and cx = cy = cz = 2(a − b)/3, so that the eigenvalues are 4λ1 = 1 + a − b, 12λ2 = 3 − a − 5b, 12λ3 = 3 + 5a + b, 12λ4 = 3 − 7a + 7b, and those of the partial tranpose 12λP T 1 = 3 + a− b, 12λP T 2 = 3 − 5a+ 5b, 12λP T 3 = 3 + 2a − 2b + √ d, 12λP T 4 = 3 + 2a − 2b − √ d with d = 25a2 − 14ab + 25b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For a = −b these states are separable for −3/14 ≤ b ≤ 1/4 and entangled for 1/4 < b ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' If the two noninteracting qubits A and B have local Hamiltonians HA = HB = σz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' we find {DA ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' ρA} = √ 2(1 − a2) 80(2 − a2) (fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b − 5bha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b) σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (24) {DB ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' ρB} = − √ 2(1 − b2) 80(2 − b2) (ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b + 5aha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b) σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' (25) where here fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b = 3Bln(λ1) − 5Bln(λ2) + 5Bln(λ3) − 3Bln(λ4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b = 3Bln(λ1) + 5Bln(λ2) − 5Bln(λ3) − 3Bln(λ4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' ha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content='b = Bln(λ1)−Bln(λ2)−Bln(λ3)+Bln(λ4) so that again the nonlinear evolution is clearly nontrivial in the sense that the local nonlinear evolution of A (B) does not depend only on ρA (ρB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' despite being no-signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' To summarize, in this Letter we prove that the SEA formalism provides a valid non-linear extension of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' To show this, we explore the definition of no-signaling for composite systems and provide generalized necessary criteria in terms of locally perceived operators, less re- strictive than the traditional criterion in terms of local density operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' Furthermore, we build on that defini- tion and show how, by construction, SEA is no-signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' For non-interacting subsystems, the traditional criterion is met for uncorrelated states, but we provide nontrivial examples of correlated states for which it is not met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFJT4oBgHgl3EQfdiwx/content/2301.11548v1.pdf'} +page_content=' RKR is grateful to the INSPIRE Fellowship program by the Department of Science and Technology, Govt.' 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Box 1627, 70211 Kuopio, Finland +˚Email: meghdoot.mozumder@uef.fi +Abstract +Diffuse optical tomography (DOT) use near-infrared light for imaging optical properties of biological tissues. Time-domain +DOT systems use pulsed lasers and measure time-varying temporal point spread function (TPSF), carrying information from +both superficial and deep layers of imaged target. +In this work, feasibility of nanosecond scale light pulses as sources for time-domain DOT is studied. Nanosecond sources +enable using relatively robust measurement setups with standard analog-to-digital converter waveform digitizers, such as +digital oscilloscopes. However, this type of systems have some properties, such as variations in source pulses and limited +temporal sampling, that could limit their usage. In this work, these different aspects and possible limitations were studied +with simulations and experiments. +Simulations showed that information carried by time-domain data of diffuse medium is on low frequencies. This enables +usage of relatively slow response time measurement electronics, and image processing using Fourier-transformed time- +domain data. Furthermore, the temporal sampling in measurements needs to be high enough to capture the TPSF, but this +rate can be achieved with standard digital oscilloscopes. It was shown that, although variations in light pulses of nanosecond +lasers are larger than those of picosecond sources, these variations do not affect significantly on image quality. Overall, the +simulations demonstrated the capability of nanosecond sources to be utilised in time-domain DOT in diffuse medium. +In this work, a prototypetime-domain DOT experimental system utilising a high-energy nanosecond laser was constructed. +The system is relatively robust consisting of a nanosecond Nd:YAG laser combined with optical parametric oscillator for +light input and optical fibres for guiding the light, and avalanche photodetector and high-bandwidth oscilloscope for TPSF +measurements. The system was used in both absolute and difference imaging of two phantoms. The experiments verified +that both absorbing and scattering objects can be reconstructed with good quality with time-domain DOT using a nanosecond +laser. +Keywords: diffuse optical tomography, time-domain, nanosecond lasers, diffusion approximation, image reconstruction +1 +Introduction +Diffuse optical tomography (DOT) is an imaging modality that uses visible red and/or near-infrared light for imaging +spatially varying optical parameters in biological tissues [1, 2, 3]. Distribution of these optical parameters provide tissue +biochemical and structural information with applications, for example, in imaging of breast cancer, monitoring neonatal +brain, functional brain imaging and pre-clinical small animal studies [4, 5, 3, 6, 7]. +Experimental DOT systems can be divided into three types depending on the light source that is used. +These are +continuous wave (CW), time-domain (TD), and frequency domain (FD) i.e. intensity modulated systems. The CW sources +enable fast data acquisition and usage of simple detectors [8]. However, CW-DOT cannot distinguish between absorption +and scattering effects unless a reference measurement to enable difference imaging is available [9, 1]. The FD-DOT systems +use radio-frequency modulated light sources for illumination, and measure amplitude attenuation and phase delay of the +transmitted light [2]. Time-domain diffuse optical tomography (TD-DOT) uses pulsed lasers for illuminating the tissues, +and the time-varying boundary exitance, i.e. temporal point spread function (TPSF), is measured. [10]. Both FD and TD +systems can be used for simultaneous absolute imaging of absorption and scattering distributions. In DOT, the TD system +are especially in the interest due to their capability to image through large thicknesses of tissue. Furthermore, in TD-DOT, +the information content of the measured TPSF is large, carrying information from both superficial and deep layers of the +tissue [2, 10]. +Generally, the TD-DOT and near-infrared spectroscopy (NIRS) systems have been based on picosecond light sources. +Wavelength-tunable light having picosecond or less duration can be generated using Ti:Sapphire and supercontinuum lasers +[11, 12, 13, 14]. These lasers operate in tens of MHz pulse repetition rates and can provide several watts (Ti:Sapphire) to +several milliwatts (supercontinuum) of power per nanometer making them suitable sources for DOT. Single wavelengths can +be generated using high repetition rate pulsed laser diodes providing tens to hundreds picosecond long pulses [15, 16, 17, 18]. +The main advantage of them over the Ti:Sapphire and supercontinuum lasers are lower cost, easier operation and smaller +This is the version of the article before peer review or editing, as submitted by an author to Measurement Science and Technology. IOP Publishing Ltd +is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at +https://iopscience.iop.org/article/10.1088/1361-6501/ac9e11/. +1 +arXiv:2301.03269v1 [physics.optics] 9 Jan 2023 + +footprint in the expense of single wavelength operation and lower output power. Measurements and light detection in +TD-DOT have been based on photon counting methods and/or time-gated detectors [15, 19, 20, 11, 21, 22, 13, 10, 18]. These +are known to be sensitive, have very high dynamic range, and they are relative fast methods for detecting light. The +sensitivity and high dynamic range enable measurements both near the light sources and over relatively large distances of +diffuse medium. Furthermore, the fast repetition rate of lasers enables detection of the entire TPSF in a reasonable time. +However, the photon counting methods require a large number of repetitions of light illuminations in order to capture the +entire TPSF. TD-DOT has been applied, for example, in imaging tissue-mimicking phantoms [19, 15, 21] and breast imaging +[23, 18]. Further, multi-channel NIRS has been applied for example in brain studies [11, 22]. +Usage of picosecond pulses with photon counting detection in TD-DOT has been motivated by good temporal resolution, +and the high repetition rate by fast acquisition time for in vivo imaging applications [10]. It has also been anticipated that a +high temporal resolution can provide a good spatial resolution, especially if early photons are detected [24, 25, 26]. On the +other hand, it has been recently shown that, in diffuse regime, information content of TD-DOT data is on low frequencies, +such that the reconstructed images are nevertheless low resolution [27]. That is, in diffuse medium, i.e. in a highly scattering +medium of size larger than several mean scattering lengths, only few frequencies are required to reconstruct absorption and +scattering with same resolution as with a full TD-DOT data [27]. +In this paper, we study feasibility of nanosecond light sources in TD-DOT. Our work is motivated by the aspiration to +enhance and develop DOT systems that could be implemented together with other imaging modalities. Such multimodality +approaches include, for example, combining DOT with EEG, MRI and ultrasound [10], and photoacoustic tomography +[28, 29, 30]. Nanosecond lasers have been previously utilised in few NIRS studies to examine, for example, fruits [31, 32], +optical phantoms [33] and brain function [34]. Further, in [16, 35] pulsed laser diodes with a pulse width between 100´400 ps +were utilised in breast and phantom imaging, respectively. However, to our knowledge, nanosecond lasers have not been +utilised in tomography similarly as picosecond lasers are used in the state-of-the-art TD-DOT systems. If light illumination +in DOT is based on nanosecond sources, measurements can be performed using relatively robust measurement setups with +standard analog-to-digital converter waveform digitizers, such as digital oscilloscopes. In that case, the full TPSF can be +measured directly without a need for time-gating or photon counting. +In this paper, feasibility of nanosecond lasers is studied with numerical simulations to evaluate different aspects and +limitations of such systems in TD-DOT. These aspects include amplitude spectrum (frequency content) of data, coarse +temporal sampling and slow response time of detection, and light source variations. The results are compared against +TD-DOT simulations with picosecond light sources. Furthermore, feasibility of an experimental TD-DOT system with a +nanosecond Nd:YAG laser source is evaluated with phantom measurements. The same laser has previously been utilised in +photoacoustic tomography experiments [36]. +The rest of the paper is organised as follows. +Theory and models for TD-DOT are described in Section 2 and the +experimental setup is described in Section 3. The numerical simulations are reported in Section 4, followed by experiments +in Section 5. Finally, the results are discussed and concluded in Section 6. +2 +Theory +In DOT measurements, visible or near-infrared light is introduced to the imaged target, and the amount of transmitted light +is measured on various positions on the boundary using light sensitive detectors. This measurement is then repeated for +multiple illumination positions leading a set of DOT data. Then, an image of the optical parameters is reconstructed from +the measured data. In this work, we consider estimation of absorption and (reduced) scattering. +2.1 +Modelling light propagation +Let us consider domain Ω Ă Rd with boundary BΩ where d is the dimension of the domain (d “ 2 or 3). Propagation of light +in a diffuse medium can be described using the diffusion approximation (DA) to the radiative transfer equation [37, 1]. The +time-domain DA together with a Robin boundary condition is +ˆ +´∇ ¨ +1 +dpµaprq ` µ1 +sprqq∇ ` µaprq ` 1 +c +B +Bt +˙ +Φpr, tq “ 0, r P Ω +(1) +Φpr, tq ` +1 +2γd +1 +dpµaprq ` µ1 +sprqqαBΦpr, tq +Bˆn +“ +# +Qpr,tq +γd , r P s +0, r P BΩzs +(2) +where Φpr, tq is the photon density at a point r and time instance t, µaprq is the absorption coefficient, and µ1 +sprq is the reduced +scattering coefficient, c is the speed of light in the medium, and Qpr, tq is the pulsed (temporal) light source at source positions +s [1]. Further, parameter γd is a dimension dependent constant that takes values γ2 = 1{π and γ3 = 1{4, α is a parameter +governing reflection at the boundary BΩ, and ˆn is an outward unit vector normal to the boundary. +2 + +The measurement data in DOT is the boundary exitance Γ`pt, rq, t “ 1, ..., T, where T is the temporal range of the output +signal at detector positions r Ă BΩ. The exitance can be solved from photon density as +Γ`pr, tq “ ´ +1 +dpµaprq ` µ1 +sprqq +BΦpr, tq +Bˆn +“ 2γd +α Φpr, tq. +(3) +Fourier transform of the time-domain DA (1)-(2), results in the frequency domain DA +ˆ +´∇ ¨ +1 +dpµaprq ` µ1 +sprqq∇ ` µaprq ` iω +c +˙ +Φpr, ωq “ 0, r P Ω +(4) +Φpr, ωq ` +1 +2γd +1 +dpµaprq ` µ1 +sprqqαBΦpr, ωq +Bˆn +“ +# +Qpr,ωq +γd +, r P s +0, r P BΩzs +(5) +where Φpr, ωq is the photon density and Qpr, ωq is the light source modulated at an angular frequency ω. Furthermore, the +frequency domain exitance can be solved from the photon density as +Γ`pr, ωq “ ´ +1 +dpµaprq ` µ1 +sprqq +BΦpr, ωq +Bˆn +“ 2γd +α Φpr, ωq. +(6) +Utilising the relation between the time domain and frequency domain light transport models, it is possible to convert the +measured TD-DOT data to frequency domain, and to use that data at one or several frequencies [27]. For light sources with +a finite temporal length, the measurable data Γ`pr, tq can be expressed as a convolution (˚) of the source Qpr, tq and exitance +due to delta source Γ` +δ pr, tq as +Γ`pr, tq “ Γ` +δ pr, tq ˚ Qpr, tq +(7) +implying that taking a Fourier transform results in product (¨) of their Fourier transforms (by convolution theorem) +F Γ`pr, tq “ F Γ` +δ pr, tq ¨ F Qpr, tq +(8) +Thus, for frequency domain computations, the Fourier transformed time-domain data needs to be divided by the Fourier +transform of the source term, and then light transport can be modelled directly in frequency domain using (4)–(6) [27]. +In this work, the solution of the DA is approximated with a finite element method [38]. The FE-approximation of the +time-domain DA (1)-((2)) is implemented as described in [27] with the time-stepping implemented with a Crank–Nicholson +scheme. Further, the frequency domain FE-computations are implemented and Fourier series approximation is utilised +similarly as described in [27, 39]. The TOAST++ software [40] is utilised in the FE-integrations. +2.2 +Image reconstruction +Let us denote the vectors of unknown absorption and (reduced) scattering coefficients as µa “ pµa,1, . . . , µa,NqT P RN and +µ1 +s “ +´ +µ1 +s,1, . . . , µ1 +s,N +¯T +P RN where N is the size of the discretisation. Further, let the measurement data vector be Γmeas P RNm +where Nm is the number of discretised data points. In TD-DOT, the data is the measured TPSF. In the FD-DOT, data typically +is the logarithm of amplitude and phase of (complex) exitance. Further, let us denote the solution of the forward model, that +maps the absorption and scattering parameters to the data, as Γ`pµa, µ1 +sq. +In DOT, typically two types of images are reconstructed: absolute and difference images. In absolute imaging, one aims +at estimating absolute values of optical parameters using a single set of measurements during which the target is assumed +to be non-varying. In difference imaging, one is interested in the change in optical parameters between two measurements. +Let us first consider absolute imaging. The image reconstruction problem can be written as a minimisation problem +p ˆµa, ˆµ1 +sq “ arg minµa,µ1s +"1 +2 +››LepΓmeas ´ Γ`pµa, µ1 +sqq +››2 ` 1 +2 +››Lµapµa ´ ηµaq +››2 ` 1 +2 +››Lµ1spµ1 +s ´ ηµ1sq +››2 +* +(9) +where Le is a weighting matrix that, from the statistical point of view, can be interpreted as the Cholesky decompostion of +the inverse of the noise covariance matrix, i.e. Γ´1 +e +“ LT +e Le [41, 42]. Further, the two latter terms in the minimised functional +(9) present prior information of the target, where ηµa and ηµ1s are the means and Lµa and Lµ1s are the Cholesky decompostion +of the covariance matrices of the prior model for absorption Γ´1 +µa “ LT +µaLµa and scattering Γ´1 +µs “ LT +µsLµs. In this work, the +minimisation problem (9) is solved using a Gauss-Newton method [43, 44]. +In difference imaging, two measurements are performed and the aim is to reconstruct the change in the optical parameters +between these measurements [1]. Consider data Γ1 +meas and Γ2 +meas of two measurements obtained from a target with optical +parameters (µ1 +a, µ11 +s ) and (µ2 +a, µ12 +s ), respectively. The aim in difference imaging is to reconstruct the change in the optical +parameters δµa “ µ2 +a ´ µ1 +a and δµ1 +s “ µ12 +s ´ µ11 +s based on the difference of the measurements as +δΓmeas “ Γ2 +meas ´ Γ1 +meas “ Jpµ1 +a, µ11 +s q p δ q µaδµ1 +s +(10) +3 + +where the J is the Jacobian of the forward model Γ` evaluated at pµ1 +a, µ11 +s q [39]. The change in optical parameters is estimated +by solving a minimisation problem +pδ ˆµa, δ ˆµ1 +sq “ arg minδµa,δµ1s +"1 +2 +››Lδe +` +δΓmeas ´ Jpµ1 +a, µ11 +s q p δ q µaδµ1 +s +˘››2 ` 1 +2 +››Lδµaδµa +››2 ` 1 +2 +››Lδµ1sδµ1 +s +››2 +* +(11) +where Lδe, Lδµa and Lδµ1s are the Cholesky decompositions of the inverse of the covariance matrices of the noise and prior +model for the optical parameter change [39]. In this work, the difference imaging minimisation problem (11), is solved using +a MATLAB built-in mldivide function. +In this work, we use Gaussian Ornstein-Uhlenbeck prrocess [45] as the prior model for absorption and scattering. The +Onrnstein-Uhlenbeck covariance function is of the form +Γµ “ σ2Ξ +(12) +where σ is the standard deviation of the prior and Ξ is a matrix which has its elements defined as +Ξij “ expp´||ri ´ rj||{ℓq +(13) +where i and j denote the row and column indices of the matrix, ri and rj denote the positions of the discretisation points, +and ℓ is the characteristic length scale of the prior describing the spatial distance that the parameter is expected to have +(significant) spatial correlation for [46, 39]. +3 +Materials and methods +3.1 +DOT system +The DOT system utilised in this work is illustrated in Fig. 1. In the system, light source is a nanosecond Nd:YAG laser +(neodymium-doped yttrium aluminum garnet) combined with optical parametric oscillator (model NT352B; Ekspla Uab, +Lithuania). The laser can operate at 670 to 2600 nm wavelength band. The pulse energy at 670 to 825 nm band is ą 90 mJ and +pulse repetition rate is 10 Hz. The pulse duration was approximately 3 ns. The laser was operated at 700 nm wavelength in +the experiments. The laser pulse energy was 0.85 mJ (average energy during the measurement session; ˘0.02 mJ standard +deviation when averaged over 100 pulses) and was measured with optical power meter with pyroelectric detector (models +StarBright and PE50BF-C, respectively; Ophir Photonics, Israel). This value was measured from the fiber end which was +connected to the target. +The laser output was directed to 3 m long main optical fiber (multimode, diameter 1 mm, numerical aperture (NA) +0.22; Ceramoptec, Germany). The light from this primary fiber was further directed to 2 m long secondary fiber (multi- +mode, diameter 1.5 mm, NA 0.39; model M134L02, Thorlabs) using two reflecting collimators (models RPC12FC-P01 and +RPC12SMA-P01; Thorlabs) that was connected to a measurement tank containing a liquid optical phantom. The space +between the two reflective collimators served as a point to estimate the input light profile and pulse-to-pulse variation. For +that purpose, laser reflection from angled glass placed between the collimators was directed to an optical fiber connected to +a fast (2 GHz) biased silicon photodetector (model DET025AFC/M; Thorlabs). Neutral density filter in front of detector was +applied to limit light level below the saturation level of the detector. +Light transmitted through the target was gathered by an optical fiber (diameter 0.6 mm, NA 0.5; model M143L02, +Thorlabs) and then focused on the centre of an amplified avalanche photodetector (model APD430A/M; Thorlabs). The +detector has temperature-compensated reverse voltage regulation and wavelength operating range from 400 nm to 1000 nm. +The detection bandwidth ranges from DC to 400 MHz (´3 dB) and rise time is approximately 1 ns. The detector gain was +set 10, which is the minimum value. +High-bandwidth oscilloscope (model WavePro 254HD; Teledyne LeCroy, NY, USA) measured the signals of both optical +detectors. The oscilloscope has measurement bandwidth up to 2.5 GHz, 12-bit dynamic range, and it was operated in 50 Ω +termination and coaxial cables. The sampling period was set to 50 ps. Signal acquisition mode was a sequence mode which +enables high accuracy measurements when the pulse repetition rate is low. +Each measurement consisted of 200 pulses. Oscilloscope was triggered using transistor-transistor logic (TTL) signal +with constant well-defined shape and amplitude from the laser Q-switch. TTL signal was used as the trigger instead of +light reflection from the glass plate since the beam profiles of laser pulses detected by the biased photodetector were found +to be varying and uneven in amplitude and shape. Trigger offset variation was found to be less than 50 ps indicating +good temporal repeatability. Time from the electrical trigger to laser pulse generation also varied. According to the laser +manufacturer, standard deviation of the jitter between the trigger and laser pulse is approximately 0.2 ns. In this work, we +measured laser pulses and studied effect of their variations by simulations. +The signal detection under computer guidance was operated using LabView software (National Instruments, TX, USA) +through an ethernet connection. After each measurement the recorded 200 source pulses and 200 measurement signals were +averaged to provide time-domain source signal and measurement data. +4 + +Figure 1: Illustration of the time-domain diffuse optical tomography system used in the study. +3.2 +Optical phantom +Optical phantom used in the study was a liquid phantom inside a black plastic cylindrical tank. The inner height, inner +diameter, and wall thickness of the tank were 155 mm, 80 mm and 5 mm, respectively. On the tank wall, at the height of +77.5 mm, 16 holes were drilled using 22.5˝ angular spacing. Eight of them had diameter of 3.2 mm and were used only as +light detection points. Another eight holes had 10 mm diameter and were used both light source and light detection points. +Liquid and the fibres were separated by a plastic membrane window (thickness 20 µm, transmission approximately 95 % +at 700 nm). The detector fibre having standard SM905 connector was directly connected and fixed to the detection points +having 3.2 mm hole. When the detection was made from 10 mm holes, custom-made steel adapter and SMA bulkhead +adapter (HASMA; Thorlabs) were used to accommodate the fiber head to hole. Similar-type adapter was used when the +source fibre was connected to 10 mm holes. Adapter positioned the secondary source fiber end at 10 mm distance from +the plastic membrane and provided 8 mm diameter exit for light beam. The estimated fluence on the phantom surface was +1.7 mJ{cm2. +Clear glass tubes (Duran®; Duran Wheaton Kimble, NJ, USA) were used as a container for the liquid optical inclusions +that provided different optical properties compared to the background liquid. The glass tubes had 130 mm height, 13.5 mm +outer diameter, and 1 mm wall thickness. The tank had a removable plastic cover and the tubes were fixed to the cover +which kept them straight and in correct position inside the liquid. The tank was filled with the background liquid up to +around 130 mm height during the measurements. +Phantom liquid was made of degassed deionized water, Intralipid (20 %, Fresenius Kabi, Sweden) and India Ink (Royal +Talens, the Netherlands). The background solution had 1 % concentration of Intralipid resulting in reduced scattering +µ1 +s “ 1 mm´1 [47, 48]. Absorption at 700 nm due to the Intralipid and water was estimated to be µa “ 0.0008 mm´1 [48]. +This was tuned with India ink to get higher absorption [49]. Ink was diluted in degassed deionized water and sonicated +to provide absorption µa “ 0.0056 mm´1 providing total absorption µa “ 0.0064 mm´1 for the background. Scattering +inclusion was made by increasing the amount of Intralipid to 5 % in background liquid and assuming linear dependence +between the scattering and Intralipid concentration. The estimated reduced scattering coefficient was therefore µ1 +s “ 5 mm´1. +The absorbing inclusion was made by increasing the concentration of India ink to give absorption µa “ 0.032 mm´1. +3.3 +Tomographic measurement +Tomographic measurement sets were made using manual source and detector fiber positioning. Three measurements were +done: one with the liquid phantom (background measurements) and two having both scattering and absorbing inclusion +tubes placed in the liquid at two different orientations. Each measurement set was done using eight different source locations +and seven detector locations per source positions. Therefore, the total number of measurements was 56 per tomography. +5 + +Reflective collimator pair +Optical fiber +and angled glass window +Laser source +(primary) +Optical fiber +(light input) +Electric trigger +signal +Optical fiber +(TTL) +(secondary to +phantom) +Detector +Ch 1 +(Biased Si) +Optical fiber +Oscilloscope +(from phantom) +Detector +Ch 2 +(APD) +Phantom +Ethernet +ComputerFigure 2: +Experimental phantom tank (left) and illustration of source and detector locations, measurement protocol and +inclusion positions in the tomography experiments (right). +The source and detector positions and inclusion orientations are illustrated in Fig. 2. +4 +Simulations +Numerical simulations were carried out to study the following aspects related to TD-DOT with nanosecond light sources. +Firstly, we wanted to study, if data obtained with nanosecond sources is lower in information such that it would lead to +different quality reconstructions than those obtained using picosecond light sources. Therefore, frequency spectra of data +simulated by nanosecond and picosecond sources were compared. Further, a comparison between absolute and difference +image reconstruction in DOT using nanosecond and picosecond light sources and one or multiple frequencies was performed. +Secondly, if DOT measurements are performed using digital oscilloscopes, temporal sampling is coarser and response time +of detection slower when compared to data given by time-correlated photon counting methods. Therefore, we studied the +effect of the temporal sampling of detection in nanosecond TD-DOT for data and image reconstruction. The slower response +time of detection corresponds to low-pass filtering in modelling, and thus it limits the frequencies that can be utilised in +image reconstruction. Thirdly, nanosecond lasers, such as the one used in our setup, can suffer large variations in light +pulse width and shape. Therefore, light pulse variations in nanosecond TD-DOT data were measured and their impact on +reconstructions were studied. The results were compared against picosecond sources with less variations. The simulations +were carried out in a Fujitsu Celcius W550 desktop workstation, with Intel®Xeon(R) W-2125 CPU @ 4.00GHzˆ8, using +MATLAB (R2017b, Mathworks, Natick, MA). +4.1 +Data simulation +In numerical simulations, a circular domain Ω Ă R2 with a radius of 25 mm was considered. The setup consisted of 16 +sources and 16 detectors. The source and detector optodes were modelled as Gaussian surface patches with 2 mm width, +located at equi-spaced angular intervals on the boundary BΩ. We studied a target with background optical parameters +µa “ 0.006 mm´1 and µ1 +s “ 1 mm´1 and with one absorbing µa “ 0.03 mm´1 and one scattering µ1 +s “ 5 mm´1 inclusion. +The chosen optical parameters roughly corresponded to those observed in biological tissues [50], and our optical phantom +described earlier in Section 3.2. For difference imaging, the reference data was simulated using constant background optical +parameter µa “ 0.006 mm´1 and µ1 +s “ 1 mm´1. +The TD data was simulated using FE-approximation of the DA (1)-(2) in a mesh with 1369 nodes and 2622 triangular +elements. The nanosecond source pulse was modelled as a Gaussian pulse with a full width at half maximum of 3 ns. For +comparison, data with a picosecond light sources was simulated. In that case, the pulse was modelled as a Gaussian pulse +with a full width at half maximum of 3 ps. The temporal discretisation for both nanosecond and picosecond data simulation +was 1 ps, and the temporal range was specified as 10000 ps. The total number of the simulated time-resolved measurements +was 12800000 (256 combination of sources and detectors, 50000 time steps). +In order to evaluate the data in the frequency domain, the time-domain data was Fourier transformed to frequency +domain. The source was deconvoluted from the data by dividing the frequency domain data by corresponding frequency +domain source term obtained through Fourier-transform [27]. Random measurement noise, that was drawn from a zero- +mean Gaussian distribution where the standard deviations were specified as 1% of the simulated noise-free frequency +6 + +BendingSource port (S1 to S8) +S4-D6 +S4-D7 +S5-D4 +S3-D7 +Detection port (D1 to D7) +S5-D5 +S6-D2 +S4-D5 +S6-D3 +S5-D3 +S1 +S5-D6 +S7-D1 +S6-D1 +S3-D6 +S6-D4 +S4-D4 +S7-D2 +S5-D2 +8 mm +3 mm +S5-D7 +μa target +S2-D7 +S6-D5 +S3-D5 +S7-D3 +S4-D3 +S5-D1 +S8-D1 +S6-D6 +80 mm +S2-D6 +S7-D4 +8 +S +S3-D4 +S8-D2 +45° or 22.5° +S4-D2 +20 mm +S1-D1 +S1-D7 +S6-D7 +S2-D5 +S7-D5 +S3-D3 +S4-D1 +S8-D3 +se +z +S1-D2 +S1-D6 +S7-D6 +S2-D4 +>x +S1-D3 +S1-D5 +S8-D4 +S3-D2 +S5 +S2-D1 +S2-D3 +y +S7-D7 +S1-D4 +S3-D1 +S8-D5 +S2-D2 +S8-D7 +S8-D6Figure 3: (a) Simulated source pulses of a 3 ns source (top row) and 3 ps source (bottom row), (b) amplitude spectra of the +source pulses, (c) simulated measurement data from a near (blue) and far (red) detector using a homogeneous target, (d) +amplitude spectra of the simulated data, and (e) amplitude spectra of the data deconvoluted with the source pulses. +domain data, was added to the data. +4.2 +Image reconstruction +In the reconstructions, a FE-mesh with 1123 nodes and 2142 elements was used. The absolute (9) and difference (11) image +reconstruction problems were solved using the Gauss-Newton method. The measurement noise was modelled using the +statistics of the simulated data, i.e. standard deviation was 1 % of the noise free data. The parameters of Ornstein-Uhlenbeck +prior were chosen such that, for absolute imaging, the prior means (ηµa, ηµ1s) were set as the background optical parameters +and the standard deviations pσµa, σµ1sq were set such that maximum target values corresponded to three standard deviations +from the background. For difference imaging, the prior means (ηδµa, ηδµ1s) were set to zero and the standard deviations +pδσµa, δσµ1sq were set such that maximum target values corresponded to five standard deviations from the background. The +characteristic length scale was set as ℓ “ 8 mm. +In addition to visual inspection, the accuracy of the estimates were evaluated by computing relative errors +Eµ “ 100% ¨ } ˆµ ´ µtarget} +}µtarget} +(14) +where µtarget are the simulated target distributions for absorption or scattering, and ˆµ are the estimated parameters interpo- +lated to the simulation grid. +4.3 +Comparison of data and reconstructions using nanosecond and picosecond sources +First, frequency content of data simulated by nanosecond and picosecond pulses and reconstructions computed from this +data were investigated. It was studied, if data obtained with nanosecond sources has lower frequency content such that it +would lead to different quality reconstructions than those obtained from data using picosecond light sources. +In order to study the frequency content of the simulated TD-DOT data using 3 ns and 3 ps light sources, we simulated +data for detector positions located close and far from the illuminating fibre. These detector positions correspond to locations +adjacent of the source fibre and on the opposite side of the simulation domain. The data was simulated with a target with +constant (background) optical parameters. The simulated source pulses and the corresponding data at near and far detectors +of nanosecond and picosecond sources are shown in Fig. 3 together with the amplitude spectra the signals. The figure also +shows the amplitude spectra of the simulated data deconvoluted with their source pulses. +As it can be seen from column (b) of Fig. 3, picosecond sources have higher frequency content than nanosecond sources. +It can also be seen (columns (d) and (e)), that data measured near the source have higher frequency content than data +measured far from the source. However, although data obtained using nanosecond and picosecond sources have different +spectra (column (d)), the information that they provide from the target is similar, which can be seen in the amplitude spectra +of the data deconvoluted with their source pulses (column (e)). It is especially notable that the frequency content of data +measured far from the source (red lines in column (e)), that carries most information on target interior, is low. This predicts +that resolution of images reconstructed from this data will be low. +7 + +(a) Source Pulse .. (b) Spectra (c) Measurements. (d) Spectra (e) Deconvoluted +pulse +5001000 +0 +10000 +0 +500 +1000 +0 +10000 +0 +500 +1000 +pulse +10000 +0 +0 +500 +1000 +0 +10000 +0 +500 +1000 +0 +500 +1000 +time (ps) +freq (MHz) +time (ps) +freq (MHz) +freq (MHz)Figure 4: Absolute imaging of absorption µa (top row) and scattering µ1 +s (bottom row) distributions using data at different +number of frequencies generated with a 3 ns light source. Columns from left to right: (a) Simulated target, and (b)-(f) +estimates obtained using one to five frequencies. Relative errors of the estimates (14) are give below each image. Statistics +of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g). +Figure 5: Absolute imaging of absorption µa (top row) and scattering µ1 +s (bottom row) distributions using data at different +number of frequencies generated with a 3 ps light source. Columns from left to right: (a) Simulated target, and (b)-(f) +estimates obtained using one to five frequencies. Relative errors of the estimates (14) are give below each image. Statistics +of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g). +Figure 6: Difference imaging of absorption µa (top row) and scattering µ1 +s (bottom row) distributions using data at different +number of frequencies generated with 3 ns light sources. Columns from left to right: (a) Simulated target difference, and +(b)-(f) estimates obtained using one to five frequencies. Relative errors of the estimates (14) are give below each image. +Statistics of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g). +The absolute and difference reconstructions using data at different number of frequencies were computed using data simu- +lated both with 3 ns and 3 ps light sources. The frequencies that were used were f “ 66.66, 133.33, 200.00, 266.66, 333.33 MHz. +Notice that the first frequencies utilised are close to those typically used in frequency domain DOT systems. The absolute +reconstructions using 3 ns light source are shown in Fig. 4 and for 3 ps light source in Fig. 5 for a different number of +frequencies. Further, the difference images using 3 ns light source are shown in Fig. 6 and for 3 ps light source in Fig. 7. +Statistics of relative errors of the reconstructions using 100 noise realisations are also shown. +8 + +(b)N =1 +(c) N. =2 +(d) N +=3 +(e) N +=4 +(f) N +=5 +(a) Target +w +w +(g) +% +Error stats +0.03 +45 +40 +35 +30 +25 +0 +Error 39% +Error 37% +Error 37% +Error 38% +Error 36% +5 +Error40% +Error37% +Error36% +Error36% +Error30% +N(b) N +=1 +(c) N +=2 +(d) N +=3 +(e) N. +=4 +(f) N +=5 +(a) Target +3 +w +0.03 +(g) % I +Error stats +45 +40 +335 +0 +Error38% +Error36% +Error37% +Error36% +Error37% +5 +4433 +2 +3 +5 +0 +Error 37% +Error36% +Error 35% +Error32% +Error31% +N(b) N =1 +(c) N =2 +(d) N +=3 +(e) N +=4 +(f) N +=5 +(a) Target +w +(g) +% +Error +0.024 +stats +→+ +80 +丰+ +60 +0 +Error 72% +Error68% +Error65% +Error69% +Error 64% +4 +80 +3 +60 +5 +Error 77% +Error 71% +Error 73% +Error 73% +Error74% +NFigure 7: Difference imaging of absorption µa (top row) and scattering µ1 +s (bottom row) distributions using data at different +number of frequencies generated with 3 ps light sources. Columns from left to right: (a) Simulated target difference, and +(b)-(f) estimates obtained using one to five frequencies. Relative errors of the estimates (14) are give below each image. +Statistics of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g). +Figure 8: +Simulated TPSF at 1 ps interval (black solid line) on a detector located opposite to the source position in +the simulation domain. Sampled TPSFs at 500 ps, 1000 ps and 2000 ps temporal resolutions (blue, red and green lines, +respectively). +The results show no apparent differences between reconstructions from data generated with nanosecond or picosecond +light sources. Furthermore, both absolute and difference imaging reconstruct the absorbing and scattering inclusion positions +with similar accuracy regardless of the number of frequencies used. On the other hand, the contrast of the images is improved +if the number of frequencies is larger than one. However, the contrast does not significantly improve if more than three +frequencies are utilised. This corresponds to our earlier findings showing that utilising multiple frequencies in imaging in +diffuse medium improves reconstructions, but only up to few frequencies when compared to full time-domain data [27]. +Based on these simulations, we chose to use data at three frequencies in later simulations of this work, as it provided +reconstructions with adequate quality and accuracy. +If DOT measurements are performed using oscilloscopes, such as in our system described in Section 3, response time +of detection is slow. This can be modelled as a low-pass filter in frequency domain computations. In practise, this limits +the frequencies in data that can be obtained and used in reconstructions. In the experimental system used in this paper, the +cut-off frequency is approximately 400 MHz, and thus the frequencies that were used were limited below this. +4.4 +Reconstructions using data sampled at different temporal resolutions +Next, the effect of low temporal sampling of data on image reconstruction was studied. Therefore, samples at different +time-intervals were taken from the simulated time-domain data. The simulated data, i.e. TPSF, on a detector located +opposite to the source on the other side of the simulation domain generated by a 3 ns light pulse and the signals sampled +with different temporal resolutions are illustrated in Fig. 8. As it can be seen, coarse sampling does not capture the original +TPSF completely, that can be expected to lead artefacts in the reconstructions. +Then, the signals with different temporal resolutions were Fourier-transformed to frequency domain where absolute and +difference reconstructions were computed using three frequencies of data, f “ 66.66, 133.33, 200.00 MHz. For difference +imaging, the simulated reference data was sampled using the same temporal sampling and transformed to frequency domain. +The absolute reconstructions from data at different temporal sampling are shown in Fig. 9, and the difference images from +9 + +X10-8 +1ps +xcitance +O500ps +1000ps +2 +2000ps +0米米 +0 +5 +10 +nanoseconds(b) N +=1 +(c) N +=2 +(d) N +=3 +(e) N +=4 +(f) N +=5 +(a) Target +0.024 +(g) % +Error stats +80 +60 +0 +Error72% +Error68% +Error 66% +Error 69% +Error64% +4 +80 +3 +60 +0 +Error 75% +Error75% +Error 72% +Error73% +Error 74% +NFigure 9: Absolute imaging of absorption µa (top row) and scattering µ1 +s (bottom row) distributions from 3 ns light source +data sampled at different temporal discretisations. Columns from left to right: (a) Simulated target, and estimates obtained +by sampling the measured TPSFs at (b) 1 ps, (c) 50 ps, (d) 500 ps, (e) 1000 ps, (f) 2000 ps and (g) 3000 ps temporal resolution. +Relative errors of the estimates (14) are given below each image. +Figure 10: +Difference imaging of absorption δµa (top row) and scattering δµ1 +s (bottom row) distributions from 3 ns light +source data sampled at different temporal discretisations. Columns from left to right: (a) Simulated target difference, and +estimates obtained by sampling the measured TPSFs at (b) 1 ps, (c) 50 ps, (d) 500 ps, (e) 1000 ps, (f) 2000 ps and (g) 3000 ps +temporal resolution. Relative errors of the estimates (14) are given below each image. +data at different temporal sampling are shown in Fig. 10. As it can be seen from Fig. 9, the absolute reconstructions obtained +from data with temporal sampling between 1 ps and 1000 ps look qualitatively similar. Further, the relative errors of the +estimates are approximately the same. However, when temporal sampling decreases even more, the reconstructions suffer +from artefacts, that are substantial at low temporal sampling (images with 2000 ps and 3000 ps temporal resolution), and +the relative errors increase. It can further be seen from Fig. 10 that difference imaging cancels out the errors caused by a +low temporal sampling to some extent. However, when the sampling is in the same level as the source pulse width, the +difference imaging cannot correct image artefacts either. +In general, the simulations demonstrate that low temporal sampling should not affect on the reconstructions as long as +the sampling is high enough to capture the temporal features of the data. In our experimental setup, the temporal sampling +is 50 ps and that should not affect the reconstructions. +4.5 +Reconstructions in the presence of light source variations +Nanosecond lasers suffer from larger variations in light pulse width and shape than picosecond lasers. Therefore, we studied +with simulations the effect of light source variations on image reconstruction. The results were compared against picosecond +sources with less variations. +For this, we measured 10 light source pulses of the laser of the experimental setup, with 50 ps sampling. Then, these +10 + +(a) Target +(b) 1ps +(c) 50ps +(d) 500ps +(e) 1000ps +(f) 2000ps +(g) 3000ps +0.03 +00 +0 +Error=34% +Error=35% +Error=34% +Error=36% +Error=283% +Error = 1884% +5 +O +Error = 32% +Error=34% +Error=35% +Error=33% +Error=84% +Error =275%(a) Target +(b) 1ps +(c) 50ps +(d) 500ps +(e) 1000ps +(f) 2000ps +(g) 3000ps +0.02 +o +Error=66% +Error = 67% +Error=68% +Error= 66% +Error = 59% +Error = 696% +4 +. +Error=73% +Error= 76% +Error =73% +Error = 72% +Error=76% +Error=560%Figure 11: Effect of nanosecond source variations on absolute and difference reconstructions. (a) Normalised source pulses +(red line) and the corresponding normalised simulated data (blue line). (b) Simulated target, and (c) absolute absorption +µa and scattering µ1 +s reconstructions. (c) Simulated target difference, and (e) difference absorption δµa and scattering δµ1 +s +reconstructions in the presence of source variations. +Figure 12: +Effect of picosecond source variations on absolute and difference reconstructions. (a) Simulated normalised +source pulses (red line) and the corresponding normalised simulated data (blue line). (b) Absolute absorption µa and +scattering µ1 +s reconstructions and (c) difference absorption δµa and scattering δµ1 +s reconstructions in the presence of source +variations. +source signals were interpolated to 0.5 ps duration and were used as light sources in simulations to simulate 10 TD-DOT +data sets. These source pulses and data were averaged similarly as in data processing of the experimental system. The +reference data for difference imaging was simulated and processed similarly. Measured source pulses are show in Fig. 11 +(a) together with the corresponding simulated data. In order to compare the reconstructions to picosecond lasers that have +less variations, a proportional source pulse variations were simulated to the picosecond light sources. For that, 1400 source +pulses were measured and their amplitude and temporal (phase) variations were calculated to be 0.03 (arbitrary units) and +0.23 ns respectively. These were then scaled to picosecond range and used to simulate 10 picosecond sources and simulate +TD-DOT data. Simulated picosecond source signals together with the corresponding simulated data are shown in Fig. 12 +(a). +Then, the simulated signals were Fourier transformed to the frequency domain, and absolute and difference images were +reconstructed similarly as earlier using data at three frequencies f “ 66.66, 133.33, 200.00 MHz. The absolute and difference +reconstructions are shown in Fig. 11 for data simulated with nanosecond light pulses and in Fig. 12 for data simulated +with picosecond light pulses. As it can be seen, the source variations do not result in any additional loss of image quality, +compared to absolute and difference images obtained without source pulse fluctuations in Figs. 4 and 6. It can also be seen +that there are no significant differences between reconstructions from nanosecond and picosecond lasers. That is, although +nanosecond lasers have larger variations, it seems that those do not affect the accuracy of the reconstructed images. +11 + +(a) Data +(b) Target +(c) Absolute +(d) Target +(e) Difference +reconstructions +reconstructions +0.024 +0.03 +0.5 +0 +10000 +Error = 36% +Error=67% +5 +ps +1 +Error=34% +Error=72%(a) Data +(b) Target +(c) Absolute +(d) Target +(e) Difference +reconstructions +reconstructions +0.024 +0.03 +0.5 +0 +10000 +Error=38% +Error= 70% +4 +5 +ps +Error=33% +Error=68%Figure 13: +(a) Raw measurement data from the homogeneous reference phantom showing the source pulse (red) and the +measurement pulse (blue) measured using an adjacent source-detector pair. (b) Mean data from 200 samples of source (red) +and measurement pulses (blue) in a truncated temporal interval. (c) Logarithm of amplitude (first half of the x-axis) and +phase (second half of the x-axis) of the Fourier transformed measurements for all source-detector configurations from a +phantom with two inclusions (red), homogeneous phantom (blue), and their difference (green). +5 +Experiments +DOT measurements were performed using the experimental system and phantoms described in Section 3. Two cylindrical +phantoms with an absorbing inclusion and a scattering inclusion approximately five times the value of the background +parameters were studied. The phantoms are illustrated in Fig. 2 together with the measurement protocol. The measurements +were performed using eight source locations (S1-S8) and seven detector locations (D1-D7). Furthermore, measurements +with a homogeneous reference phantom were made. The reference data was utilised in difference imaging. In addition, it +was used to provide a computational calibration measurement for absolute imaging. +A raw measurement signal collected from the homogeneous reference phantom on a detector adjacent to a source is +shown in Fig. 13 (a). The measurement data was averaged over 200 the samples and truncated to a temporal window of the +measurement pulse width as shown in Fig. 13 (b). Then, the data was Fourier-transformed to the frequency domain and +deconvoluted with a source by dividing the Fourier-transformed data with the corresponding Fourier-transformed source +pulse. The Fourier transformed data at frequency f “ 159.36 MHz for all source-detector combinations is shown in Fig. 13 +(c). +The difference and absolute reconstructions were computed from frequency domain data at frequency f “ 159.36 MHz. +In reconstructions, a 2D computation domain was considered. The domain was discretised using 1369 nodes and 2622 +elements. The absolute reconstructions were computed by minimising (9) and the difference reconstructions were computed +by minimising (11). The minimisation problems were solved using Gauss-Newton method similarly as for simulated data. +For absolute imaging, the measurements with inclusions were precalibrated using the following procedure. Calibration +coefficients for individual source-detector pairs were computed by taking the difference between the experimental refer- +ence measurements and simulated reference measurements. Then, this difference was subtracted from the experimental +measurements with inclusions, and the subtracted measurements were considered as calibrated data. +Reconstructed absolute and difference images for the two phantoms are shown in Fig. +14. +As it can be seen, the +location of absorption and scattering inclusions can be distinguished both using absolute and difference imaging. Also the +difference in inclusion positions between the two phantoms is clearly visible both in absolute and difference imaging. The +absorption images show some cross-talk from the scattering inclusion, that is especially evident in the difference images, +but the magnitude of the cross-talk is very low. The scattering images, on the other hand, have more artefacts both in +absolute and difference imaging. The difference imaging shows less artifacts than absorption images. That is typical in +DOT since difference imaging compensates both measurement and modelling errors. Overall, all reconstructions can be +regarded as good quality DOT images. However, the estimated inclusion values do not reach the correct absolute values of +the inclusions. This can be due to, for example, relatively low number of measurement positions that are all located on a +single plane of the 3D object. +6 +Discussion and conclusions +In this work, feasibility of utilising nanosecond sources in TD-DOT was investigated. +Different aspects and possible +limitations of TD-DOT systems with nanosecond sources were studied with simulations and experiments. +First, frequency content of time-domain DOT data simulated with nanosecond light sources was studied and compared +against picosecond source systems. The simulations verified the previous findings [27] that information of diffuse medium is +12 + +(a)Rawdata +【b)Truncated(mean)data +()Fouriertransformed +0.8 +0.8 +0 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0 +.2 +(0) +0 +50 +100 +150 +0 +10 +0 +50 +100 +Time (ns) +Time (ns)Figure 14: (a) Reconstructed absolute absorption µa (first row) and scattering µ1 +s (second row) images for the two phantoms +(first and second column). (b) Reconstructed difference absorption δµa (first row) and scattering δµ1 +s (second row) images +for the two phantoms (third and fourth column). +on low frequencies. This enables image reconstruction using Fourier-transformed time-domain data using few frequencies. +Furthermore, it enables usage of measurement electronics with slower response time. Second, effect of temporal sampling +was studied. The simulations showed that temporal sampling needs to be high enough to capture the TPSF. This sampling +can be achieved with standard digital oscilloscopes. Third, nanosecond lasers can suffer from light pulse variations. It was +shown that, although these variations are large, they do not affect significantly on image quality. Overall, the simulations +demonstrated the capability of nanosecond sources to be utilised in TD-DOT in diffuse medium. +Then, a prototype TD-DOT experimental system utilising a high-energy nanosecond source was constructed. +The +system consisted of a nanosecond Nd:YAG laser combined with optical parametric oscillator for light input, optical fibres +and collimators for guiding the light, and avalanche photodetector and high-bandwidth oscilloscope for measurements. +The system is relatively robust. For example, detector is temperature-compensated, it does not require cooling and can +be operated in normal room lighting without saturation. The pros of the system include broad wavelength tuning range +for multiwavelength imaging as well as high energy per wavelength and pulse, that would enable directing the light to +multiple fibres. Further, relative impacts of pulse temporal spreading in long optical fibres, inaccuracies on fibre lengths, +or changes in environmental conditions such as temperature and vibrations are smaller for nanosecond pulses and could +provide robustness and stability for the approach. Furthermore, a nanosecond laser and a established digitizer based signal +detection enable compatibility of the system with other techniques such as ultrasound imaging (signal waveform detection) +and photoacoustic tomography (signal waveform detection and laser source). The cons of the system include poor laser +pulse stability, that was not an issue in the experiments of the study but can be significant in some other applications, and +the low pulse repetition frequency of the laser. Also the dynamic range and sensitivity of the system may have limitations +in larger targets, which would require more research. +The DOT system was used in both absolute and difference imaging of two phantoms. It was shown that both absorbing +and scattering objects could be reconstructed. The locations of the inclusions were found, and the cross-talk between the +absorbing and scattering targets was low. The reconstructions could be improved, for example, by adding more measurement +layers to the phantom and extending modelling to 3D. +The simulations and experiments of this work were the first study demonstrating usage of nanosecond sources in TD- +DOT. The nanosecond system can be utilised in diffuse medium, that is in highly scattering medium when the imaged +target size is larger than multiple free scattering lengths. However, its performance, for example, in dilute medium where +photons travel faster from source to detector or with measurement setups with short source-detector distances were not +studied. Overall, developments of DOT systems are guided by potential applications and their different requirements on, +for example, imaging depth, speed, invasiveness, scalability and multimodality [10]. Therefore, further work is needed for +the development of the proposed setup, and finding the applications where it could be seen as most beneficial. The system +could be further developed by building source energy specific light detection utilising, for example, neutral density filters +or laser beam attenuators. Sensitivity of the system could be increased using large diameter detection fibre bundles. +13 + +(a) Absolute reconstructions (b) Absolute reconstructions +(c) Difference reconstructions(d) Difference reconstructions +phantom 1 +0.0084 +phantom 2 +0.0080 +phantom 1 +0.0012 +phantom 2 +0.001 +0.0054 +0.0057 +1.19 +1.12 +0.11 +0.11 +0.78 +0.90Acknowledgments +This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 +research and innovation programme (grant agreement No 101001417- QUANTOM). The work has been supported by the +Academy of Finland (projects 314411, 336799 Centre of Excellence in Inverse Modeling and Imaging, and 320166 the Flagship +Program Photonics Research and Innovation) and Finnish Cultural Foundation (project 00200746). +References +References +[1] Arridge SR. Optical tomography in medical imaging. 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Phys Med Biol. 2013;58(11):R37. +16 + diff --git a/udE1T4oBgHgl3EQfkASI/content/tmp_files/load_file.txt b/udE1T4oBgHgl3EQfkASI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b38a6681faf132add3dbdf36e681d481692bc3a --- /dev/null +++ b/udE1T4oBgHgl3EQfkASI/content/tmp_files/load_file.txt @@ -0,0 +1,793 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf,len=792 +page_content='Utilising nanosecond sources in diffuse optical tomography Meghdoot Mozumder˚,1, Jarkko Leskinen1, Tanja Tarvainen1 1Department of Applied Physics, University of Eastern Finland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Box 1627, 70211 Kuopio, Finland ˚Email: meghdoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='mozumder@uef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='fi Abstract Diffuse optical tomography (DOT) use near-infrared light for imaging optical properties of biological tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Time-domain DOT systems use pulsed lasers and measure time-varying temporal point spread function (TPSF), carrying information from both superficial and deep layers of imaged target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, feasibility of nanosecond scale light pulses as sources for time-domain DOT is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Nanosecond sources enable using relatively robust measurement setups with standard analog-to-digital converter waveform digitizers, such as digital oscilloscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, this type of systems have some properties, such as variations in source pulses and limited temporal sampling, that could limit their usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, these different aspects and possible limitations were studied with simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Simulations showed that information carried by time-domain data of diffuse medium is on low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This enables usage of relatively slow response time measurement electronics, and image processing using Fourier-transformed time- domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, the temporal sampling in measurements needs to be high enough to capture the TPSF, but this rate can be achieved with standard digital oscilloscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It was shown that, although variations in light pulses of nanosecond lasers are larger than those of picosecond sources, these variations do not affect significantly on image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Overall, the simulations demonstrated the capability of nanosecond sources to be utilised in time-domain DOT in diffuse medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, a prototypetime-domain DOT experimental system utilising a high-energy nanosecond laser was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The system is relatively robust consisting of a nanosecond Nd:YAG laser combined with optical parametric oscillator for light input and optical fibres for guiding the light, and avalanche photodetector and high-bandwidth oscilloscope for TPSF measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The system was used in both absolute and difference imaging of two phantoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The experiments verified that both absorbing and scattering objects can be reconstructed with good quality with time-domain DOT using a nanosecond laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Keywords: diffuse optical tomography, time-domain, nanosecond lasers, diffusion approximation, image reconstruction 1 Introduction Diffuse optical tomography (DOT) is an imaging modality that uses visible red and/or near-infrared light for imaging spatially varying optical parameters in biological tissues [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Distribution of these optical parameters provide tissue biochemical and structural information with applications, for example, in imaging of breast cancer, monitoring neonatal brain, functional brain imaging and pre-clinical small animal studies [4, 5, 3, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Experimental DOT systems can be divided into three types depending on the light source that is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These are continuous wave (CW), time-domain (TD), and frequency domain (FD) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' intensity modulated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The CW sources enable fast data acquisition and usage of simple detectors [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, CW-DOT cannot distinguish between absorption and scattering effects unless a reference measurement to enable difference imaging is available [9, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The FD-DOT systems use radio-frequency modulated light sources for illumination, and measure amplitude attenuation and phase delay of the transmitted light [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Time-domain diffuse optical tomography (TD-DOT) uses pulsed lasers for illuminating the tissues, and the time-varying boundary exitance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' temporal point spread function (TPSF), is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Both FD and TD systems can be used for simultaneous absolute imaging of absorption and scattering distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In DOT, the TD system are especially in the interest due to their capability to image through large thicknesses of tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, in TD-DOT, the information content of the measured TPSF is large, carrying information from both superficial and deep layers of the tissue [2, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Generally, the TD-DOT and near-infrared spectroscopy (NIRS) systems have been based on picosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Wavelength-tunable light having picosecond or less duration can be generated using Ti:Sapphire and supercontinuum lasers [11, 12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These lasers operate in tens of MHz pulse repetition rates and can provide several watts (Ti:Sapphire) to several milliwatts (supercontinuum) of power per nanometer making them suitable sources for DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Single wavelengths can be generated using high repetition rate pulsed laser diodes providing tens to hundreds picosecond long pulses [15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The main advantage of them over the Ti:Sapphire and supercontinuum lasers are lower cost, easier operation and smaller This is the version of the article before peer review or editing, as submitted by an author to Measurement Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The Version of Record is available online at https://iopscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='org/article/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='1088/1361-6501/ac9e11/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03269v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='optics] 9 Jan 2023 footprint in the expense of single wavelength operation and lower output power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Measurements and light detection in TD-DOT have been based on photon counting methods and/or time-gated detectors [15, 19, 20, 11, 21, 22, 13, 10, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These are known to be sensitive, have very high dynamic range, and they are relative fast methods for detecting light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The sensitivity and high dynamic range enable measurements both near the light sources and over relatively large distances of diffuse medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, the fast repetition rate of lasers enables detection of the entire TPSF in a reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, the photon counting methods require a large number of repetitions of light illuminations in order to capture the entire TPSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' TD-DOT has been applied, for example, in imaging tissue-mimicking phantoms [19, 15, 21] and breast imaging [23, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, multi-channel NIRS has been applied for example in brain studies [11, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Usage of picosecond pulses with photon counting detection in TD-DOT has been motivated by good temporal resolution, and the high repetition rate by fast acquisition time for in vivo imaging applications [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It has also been anticipated that a high temporal resolution can provide a good spatial resolution, especially if early photons are detected [24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' On the other hand, it has been recently shown that, in diffuse regime, information content of TD-DOT data is on low frequencies, such that the reconstructed images are nevertheless low resolution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' That is, in diffuse medium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' in a highly scattering medium of size larger than several mean scattering lengths, only few frequencies are required to reconstruct absorption and scattering with same resolution as with a full TD-DOT data [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this paper, we study feasibility of nanosecond light sources in TD-DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Our work is motivated by the aspiration to enhance and develop DOT systems that could be implemented together with other imaging modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Such multimodality approaches include, for example, combining DOT with EEG, MRI and ultrasound [10], and photoacoustic tomography [28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Nanosecond lasers have been previously utilised in few NIRS studies to examine, for example, fruits [31, 32], optical phantoms [33] and brain function [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, in [16, 35] pulsed laser diodes with a pulse width between 100´400 ps were utilised in breast and phantom imaging, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, to our knowledge, nanosecond lasers have not been utilised in tomography similarly as picosecond lasers are used in the state-of-the-art TD-DOT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' If light illumination in DOT is based on nanosecond sources, measurements can be performed using relatively robust measurement setups with standard analog-to-digital converter waveform digitizers, such as digital oscilloscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In that case, the full TPSF can be measured directly without a need for time-gating or photon counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this paper, feasibility of nanosecond lasers is studied with numerical simulations to evaluate different aspects and limitations of such systems in TD-DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These aspects include amplitude spectrum (frequency content) of data, coarse temporal sampling and slow response time of detection, and light source variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The results are compared against TD-DOT simulations with picosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, feasibility of an experimental TD-DOT system with a nanosecond Nd:YAG laser source is evaluated with phantom measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The same laser has previously been utilised in photoacoustic tomography experiments [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Theory and models for TD-DOT are described in Section 2 and the experimental setup is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The numerical simulations are reported in Section 4, followed by experiments in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Finally, the results are discussed and concluded in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 2 Theory In DOT measurements, visible or near-infrared light is introduced to the imaged target, and the amount of transmitted light is measured on various positions on the boundary using light sensitive detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This measurement is then repeated for multiple illumination positions leading a set of DOT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, an image of the optical parameters is reconstructed from the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, we consider estimation of absorption and (reduced) scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='1 Modelling light propagation Let us consider domain Ω Ă Rd with boundary BΩ where d is the dimension of the domain (d “ 2 or 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Propagation of light in a diffuse medium can be described using the diffusion approximation (DA) to the radiative transfer equation [37, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The time-domain DA together with a Robin boundary condition is ˆ ´∇ ¨ 1 dpµaprq ` µ1 sprqq∇ ` µaprq ` 1 c B Bt ˙ Φpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq “ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' r P Ω (1) Φpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq ` 1 2γd 1 dpµaprq ` µ1 sprqqαBΦpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq Bˆn “ # Qpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='tq γd ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' r P s 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' r P BΩzs (2) where Φpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq is the photon density at a point r and time instance t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' µaprq is the absorption coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' and µ1 sprq is the reduced scattering coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' c is the speed of light in the medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' and Qpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq is the pulsed (temporal) light source at source positions s [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, parameter γd is a dimension dependent constant that takes values γ2 = 1{π and γ3 = 1{4, α is a parameter governing reflection at the boundary BΩ, and ˆn is an outward unit vector normal to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 2 The measurement data in DOT is the boundary exitance Γ`pt, rq, t “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=', T, where T is the temporal range of the output signal at detector positions r Ă BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The exitance can be solved from photon density as Γ`pr, tq “ ´ 1 dpµaprq ` µ1 sprqq BΦpr, tq Bˆn “ 2γd α Φpr, tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (3) Fourier transform of the time-domain DA (1)-(2), results in the frequency domain DA ˆ ´∇ ¨ 1 dpµaprq ` µ1 sprqq∇ ` µaprq ` iω c ˙ Φpr, ωq “ 0, r P Ω (4) Φpr, ωq ` 1 2γd 1 dpµaprq ` µ1 sprqqαBΦpr, ωq Bˆn “ # Qpr,ωq γd , r P s 0, r P BΩzs (5) where Φpr, ωq is the photon density and Qpr, ωq is the light source modulated at an angular frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, the frequency domain exitance can be solved from the photon density as Γ`pr, ωq “ ´ 1 dpµaprq ` µ1 sprqq BΦpr, ωq Bˆn “ 2γd α Φpr, ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (6) Utilising the relation between the time domain and frequency domain light transport models, it is possible to convert the measured TD-DOT data to frequency domain, and to use that data at one or several frequencies [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For light sources with a finite temporal length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' the measurable data Γ`pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq can be expressed as a convolution (˚) of the source Qpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq and exitance due to delta source Γ` δ pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq as Γ`pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq “ Γ` δ pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq ˚ Qpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq (7) implying that taking a Fourier transform results in product (¨) of their Fourier transforms (by convolution theorem) F Γ`pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq “ F Γ` δ pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq ¨ F Qpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' tq (8) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' for frequency domain computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' the Fourier transformed time-domain data needs to be divided by the Fourier transform of the source term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' and then light transport can be modelled directly in frequency domain using (4)–(6) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, the solution of the DA is approximated with a finite element method [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The FE-approximation of the time-domain DA (1)-((2)) is implemented as described in [27] with the time-stepping implemented with a Crank–Nicholson scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, the frequency domain FE-computations are implemented and Fourier series approximation is utilised similarly as described in [27, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The TOAST++ software [40] is utilised in the FE-integrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 Image reconstruction Let us denote the vectors of unknown absorption and (reduced) scattering coefficients as µa “ pµa,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' , µa,NqT P RN and µ1 s “ ´ µ1 s,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' , µ1 s,N ¯T P RN where N is the size of the discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, let the measurement data vector be Γmeas P RNm where Nm is the number of discretised data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In TD-DOT, the data is the measured TPSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In the FD-DOT, data typically is the logarithm of amplitude and phase of (complex) exitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, let us denote the solution of the forward model, that maps the absorption and scattering parameters to the data, as Γ`pµa, µ1 sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In DOT, typically two types of images are reconstructed: absolute and difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In absolute imaging, one aims at estimating absolute values of optical parameters using a single set of measurements during which the target is assumed to be non-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In difference imaging, one is interested in the change in optical parameters between two measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Let us first consider absolute imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The image reconstruction problem can be written as a minimisation problem p ˆµa, ˆµ1 sq “ arg minµa,µ1s "1 2 ››LepΓmeas ´ Γ`pµa, µ1 sqq ››2 ` 1 2 ››Lµapµa ´ ηµaq ››2 ` 1 2 ››Lµ1spµ1 s ´ ηµ1sq ››2 (9) where Le is a weighting matrix that, from the statistical point of view, can be interpreted as the Cholesky decompostion of the inverse of the noise covariance matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Γ´1 e “ LT e Le [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, the two latter terms in the minimised functional (9) present prior information of the target, where ηµa and ηµ1s are the means and Lµa and Lµ1s are the Cholesky decompostion of the covariance matrices of the prior model for absorption Γ´1 µa “ LT µaLµa and scattering Γ´1 µs “ LT µsLµs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, the minimisation problem (9) is solved using a Gauss-Newton method [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In difference imaging, two measurements are performed and the aim is to reconstruct the change in the optical parameters between these measurements [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Consider data Γ1 meas and Γ2 meas of two measurements obtained from a target with optical parameters (µ1 a, µ11 s ) and (µ2 a, µ12 s ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The aim in difference imaging is to reconstruct the change in the optical parameters δµa “ µ2 a ´ µ1 a and δµ1 s “ µ12 s ´ µ11 s based on the difference of the measurements as δΓmeas “ Γ2 meas ´ Γ1 meas “ Jpµ1 a, µ11 s q p δ q µaδµ1 s (10) 3 where the J is the Jacobian of the forward model Γ` evaluated at pµ1 a, µ11 s q [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The change in optical parameters is estimated by solving a minimisation problem pδ ˆµa, δ ˆµ1 sq “ arg minδµa,δµ1s "1 2 ››Lδe ` δΓmeas ´ Jpµ1 a, µ11 s q p δ q µaδµ1 s ˘››2 ` 1 2 ››Lδµaδµa ››2 ` 1 2 ››Lδµ1sδµ1 s ››2 (11) where Lδe, Lδµa and Lδµ1s are the Cholesky decompositions of the inverse of the covariance matrices of the noise and prior model for the optical parameter change [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, the difference imaging minimisation problem (11), is solved using a MATLAB built-in mldivide function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, we use Gaussian Ornstein-Uhlenbeck prrocess [45] as the prior model for absorption and scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The Onrnstein-Uhlenbeck covariance function is of the form Γµ “ σ2Ξ (12) where σ is the standard deviation of the prior and Ξ is a matrix which has its elements defined as Ξij “ expp´||ri ´ rj||{ℓq (13) where i and j denote the row and column indices of the matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' ri and rj denote the positions of the discretisation points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' and ℓ is the characteristic length scale of the prior describing the spatial distance that the parameter is expected to have (significant) spatial correlation for [46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 3 Materials and methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='1 DOT system The DOT system utilised in this work is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In the system, light source is a nanosecond Nd:YAG laser (neodymium-doped yttrium aluminum garnet) combined with optical parametric oscillator (model NT352B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Ekspla Uab, Lithuania).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The laser can operate at 670 to 2600 nm wavelength band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The pulse energy at 670 to 825 nm band is ą 90 mJ and pulse repetition rate is 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The pulse duration was approximately 3 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The laser was operated at 700 nm wavelength in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The laser pulse energy was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='85 mJ (average energy during the measurement session;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' ˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='02 mJ standard deviation when averaged over 100 pulses) and was measured with optical power meter with pyroelectric detector (models StarBright and PE50BF-C, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Ophir Photonics, Israel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This value was measured from the fiber end which was connected to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The laser output was directed to 3 m long main optical fiber (multimode, diameter 1 mm, numerical aperture (NA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Ceramoptec, Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The light from this primary fiber was further directed to 2 m long secondary fiber (multi- mode, diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 mm, NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' model M134L02, Thorlabs) using two reflecting collimators (models RPC12FC-P01 and RPC12SMA-P01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Thorlabs) that was connected to a measurement tank containing a liquid optical phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The space between the two reflective collimators served as a point to estimate the input light profile and pulse-to-pulse variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For that purpose, laser reflection from angled glass placed between the collimators was directed to an optical fiber connected to a fast (2 GHz) biased silicon photodetector (model DET025AFC/M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Thorlabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Neutral density filter in front of detector was applied to limit light level below the saturation level of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Light transmitted through the target was gathered by an optical fiber (diameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='6 mm, NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' model M143L02, Thorlabs) and then focused on the centre of an amplified avalanche photodetector (model APD430A/M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Thorlabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The detector has temperature-compensated reverse voltage regulation and wavelength operating range from 400 nm to 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The detection bandwidth ranges from DC to 400 MHz (´3 dB) and rise time is approximately 1 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The detector gain was set 10, which is the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' High-bandwidth oscilloscope (model WavePro 254HD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Teledyne LeCroy, NY, USA) measured the signals of both optical detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The oscilloscope has measurement bandwidth up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 GHz, 12-bit dynamic range, and it was operated in 50 Ω termination and coaxial cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The sampling period was set to 50 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Signal acquisition mode was a sequence mode which enables high accuracy measurements when the pulse repetition rate is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Each measurement consisted of 200 pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Oscilloscope was triggered using transistor-transistor logic (TTL) signal with constant well-defined shape and amplitude from the laser Q-switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' TTL signal was used as the trigger instead of light reflection from the glass plate since the beam profiles of laser pulses detected by the biased photodetector were found to be varying and uneven in amplitude and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Trigger offset variation was found to be less than 50 ps indicating good temporal repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Time from the electrical trigger to laser pulse generation also varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' According to the laser manufacturer, standard deviation of the jitter between the trigger and laser pulse is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In this work, we measured laser pulses and studied effect of their variations by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The signal detection under computer guidance was operated using LabView software (National Instruments, TX, USA) through an ethernet connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' After each measurement the recorded 200 source pulses and 200 measurement signals were averaged to provide time-domain source signal and measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4 Figure 1: Illustration of the time-domain diffuse optical tomography system used in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 Optical phantom Optical phantom used in the study was a liquid phantom inside a black plastic cylindrical tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The inner height, inner diameter, and wall thickness of the tank were 155 mm, 80 mm and 5 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' On the tank wall, at the height of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 mm, 16 holes were drilled using 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5˝ angular spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Eight of them had diameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 mm and were used only as light detection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Another eight holes had 10 mm diameter and were used both light source and light detection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Liquid and the fibres were separated by a plastic membrane window (thickness 20 µm, transmission approximately 95 % at 700 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The detector fibre having standard SM905 connector was directly connected and fixed to the detection points having 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 mm hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' When the detection was made from 10 mm holes, custom-made steel adapter and SMA bulkhead adapter (HASMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Thorlabs) were used to accommodate the fiber head to hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Similar-type adapter was used when the source fibre was connected to 10 mm holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Adapter positioned the secondary source fiber end at 10 mm distance from the plastic membrane and provided 8 mm diameter exit for light beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The estimated fluence on the phantom surface was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='7 mJ{cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Clear glass tubes (Duran®;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Duran Wheaton Kimble, NJ, USA) were used as a container for the liquid optical inclusions that provided different optical properties compared to the background liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The glass tubes had 130 mm height, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 mm outer diameter, and 1 mm wall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The tank had a removable plastic cover and the tubes were fixed to the cover which kept them straight and in correct position inside the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The tank was filled with the background liquid up to around 130 mm height during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Phantom liquid was made of degassed deionized water, Intralipid (20 %, Fresenius Kabi, Sweden) and India Ink (Royal Talens, the Netherlands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The background solution had 1 % concentration of Intralipid resulting in reduced scattering µ1 s “ 1 mm´1 [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Absorption at 700 nm due to the Intralipid and water was estimated to be µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0008 mm´1 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This was tuned with India ink to get higher absorption [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Ink was diluted in degassed deionized water and sonicated to provide absorption µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0056 mm´1 providing total absorption µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0064 mm´1 for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Scattering inclusion was made by increasing the amount of Intralipid to 5 % in background liquid and assuming linear dependence between the scattering and Intralipid concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The estimated reduced scattering coefficient was therefore µ1 s “ 5 mm´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absorbing inclusion was made by increasing the concentration of India ink to give absorption µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='032 mm´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='3 Tomographic measurement Tomographic measurement sets were made using manual source and detector fiber positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Three measurements were done: one with the liquid phantom (background measurements) and two having both scattering and absorbing inclusion tubes placed in the liquid at two different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Each measurement set was done using eight different source locations and seven detector locations per source positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, the total number of measurements was 56 per tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 5 Reflective collimator pair Optical fiber and angled glass window Laser source (primary) Optical fiber (light input) Electric trigger signal Optical fiber (TTL) (secondary to phantom) Detector Ch 1 (Biased Si) Optical fiber Oscilloscope (from phantom) Detector Ch 2 (APD) Phantom Ethernet ComputerFigure 2: Experimental phantom tank (left) and illustration of source and detector locations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' measurement protocol and inclusion positions in the tomography experiments (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The source and detector positions and inclusion orientations are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4 Simulations Numerical simulations were carried out to study the following aspects related to TD-DOT with nanosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Firstly, we wanted to study, if data obtained with nanosecond sources is lower in information such that it would lead to different quality reconstructions than those obtained using picosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, frequency spectra of data simulated by nanosecond and picosecond sources were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, a comparison between absolute and difference image reconstruction in DOT using nanosecond and picosecond light sources and one or multiple frequencies was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Secondly, if DOT measurements are performed using digital oscilloscopes, temporal sampling is coarser and response time of detection slower when compared to data given by time-correlated photon counting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, we studied the effect of the temporal sampling of detection in nanosecond TD-DOT for data and image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The slower response time of detection corresponds to low-pass filtering in modelling, and thus it limits the frequencies that can be utilised in image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Thirdly, nanosecond lasers, such as the one used in our setup, can suffer large variations in light pulse width and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, light pulse variations in nanosecond TD-DOT data were measured and their impact on reconstructions were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The results were compared against picosecond sources with less variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The simulations were carried out in a Fujitsu Celcius W550 desktop workstation, with Intel®Xeon(R) W-2125 CPU @ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='00GHzˆ8, using MATLAB (R2017b, Mathworks, Natick, MA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='1 Data simulation In numerical simulations, a circular domain Ω Ă R2 with a radius of 25 mm was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The setup consisted of 16 sources and 16 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The source and detector optodes were modelled as Gaussian surface patches with 2 mm width, located at equi-spaced angular intervals on the boundary BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' We studied a target with background optical parameters µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='006 mm´1 and µ1 s “ 1 mm´1 and with one absorbing µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 mm´1 and one scattering µ1 s “ 5 mm´1 inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The chosen optical parameters roughly corresponded to those observed in biological tissues [50], and our optical phantom described earlier in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For difference imaging, the reference data was simulated using constant background optical parameter µa “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='006 mm´1 and µ1 s “ 1 mm´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The TD data was simulated using FE-approximation of the DA (1)-(2) in a mesh with 1369 nodes and 2622 triangular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The nanosecond source pulse was modelled as a Gaussian pulse with a full width at half maximum of 3 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For comparison, data with a picosecond light sources was simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In that case, the pulse was modelled as a Gaussian pulse with a full width at half maximum of 3 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The temporal discretisation for both nanosecond and picosecond data simulation was 1 ps, and the temporal range was specified as 10000 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The total number of the simulated time-resolved measurements was 12800000 (256 combination of sources and detectors, 50000 time steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In order to evaluate the data in the frequency domain, the time-domain data was Fourier transformed to frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The source was deconvoluted from the data by dividing the frequency domain data by corresponding frequency domain source term obtained through Fourier-transform [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Random measurement noise, that was drawn from a zero- mean Gaussian distribution where the standard deviations were specified as 1% of the simulated noise-free frequency 6 BendingSource port (S1 to S8) S4-D6 S4-D7 S5-D4 S3-D7 Detection port (D1 to D7) S5-D5 S6-D2 S4-D5 S6-D3 S5-D3 S1 S5-D6 S7-D1 S6-D1 S3-D6 S6-D4 S4-D4 S7-D2 S5-D2 8 mm 3 mm S5-D7 μa target S2-D7 S6-D5 S3-D5 S7-D3 S4-D3 S5-D1 S8-D1 S6-D6 80 mm S2-D6 S7-D4 8 S S3-D4 S8-D2 45° or 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5° S4-D2 20 mm S1-D1 S1-D7 S6-D7 S2-D5 S7-D5 S3-D3 S4-D1 S8-D3 se z S1-D2 S1-D6 S7-D6 S2-D4 >x S1-D3 S1-D5 S8-D4 S3-D2 S5 S2-D1 S2-D3 y S7-D7 S1-D4 S3-D1 S8-D5 S2-D2 S8-D7 S8-D6Figure 3: (a) Simulated source pulses of a 3 ns source (top row) and 3 ps source (bottom row),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (b) amplitude spectra of the source pulses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (c) simulated measurement data from a near (blue) and far (red) detector using a homogeneous target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (d) amplitude spectra of the simulated data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' and (e) amplitude spectra of the data deconvoluted with the source pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' domain data, was added to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 Image reconstruction In the reconstructions, a FE-mesh with 1123 nodes and 2142 elements was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absolute (9) and difference (11) image reconstruction problems were solved using the Gauss-Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The measurement noise was modelled using the statistics of the simulated data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' standard deviation was 1 % of the noise free data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The parameters of Ornstein-Uhlenbeck prior were chosen such that, for absolute imaging, the prior means (ηµa, ηµ1s) were set as the background optical parameters and the standard deviations pσµa, σµ1sq were set such that maximum target values corresponded to three standard deviations from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For difference imaging, the prior means (ηδµa, ηδµ1s) were set to zero and the standard deviations pδσµa, δσµ1sq were set such that maximum target values corresponded to five standard deviations from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The characteristic length scale was set as ℓ “ 8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In addition to visual inspection, the accuracy of the estimates were evaluated by computing relative errors Eµ “ 100% ¨ } ˆµ ´ µtarget} }µtarget} (14) where µtarget are the simulated target distributions for absorption or scattering, and ˆµ are the estimated parameters interpo- lated to the simulation grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='3 Comparison of data and reconstructions using nanosecond and picosecond sources First, frequency content of data simulated by nanosecond and picosecond pulses and reconstructions computed from this data were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It was studied, if data obtained with nanosecond sources has lower frequency content such that it would lead to different quality reconstructions than those obtained from data using picosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In order to study the frequency content of the simulated TD-DOT data using 3 ns and 3 ps light sources, we simulated data for detector positions located close and far from the illuminating fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These detector positions correspond to locations adjacent of the source fibre and on the opposite side of the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The data was simulated with a target with constant (background) optical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The simulated source pulses and the corresponding data at near and far detectors of nanosecond and picosecond sources are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 3 together with the amplitude spectra the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The figure also shows the amplitude spectra of the simulated data deconvoluted with their source pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' As it can be seen from column (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 3, picosecond sources have higher frequency content than nanosecond sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It can also be seen (columns (d) and (e)), that data measured near the source have higher frequency content than data measured far from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, although data obtained using nanosecond and picosecond sources have different spectra (column (d)), the information that they provide from the target is similar, which can be seen in the amplitude spectra of the data deconvoluted with their source pulses (column (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It is especially notable that the frequency content of data measured far from the source (red lines in column (e)), that carries most information on target interior, is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This predicts that resolution of images reconstructed from this data will be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 7 (a) Source Pulse .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='. (b) Spectra (c) Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (d) Spectra (e) Deconvoluted pulse 5001000 0 10000 0 500 1000 0 10000 0 500 1000 pulse 10000 0 0 500 1000 0 10000 0 500 1000 0 500 1000 time (ps) freq (MHz) time (ps) freq (MHz) freq (MHz)Figure 4: Absolute imaging of absorption µa (top row) and scattering µ1 s (bottom row) distributions using data at different number of frequencies generated with a 3 ns light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Columns from left to right: (a) Simulated target, and (b)-(f) estimates obtained using one to five frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Relative errors of the estimates (14) are give below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Statistics of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Figure 5: Absolute imaging of absorption µa (top row) and scattering µ1 s (bottom row) distributions using data at different number of frequencies generated with a 3 ps light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Columns from left to right: (a) Simulated target, and (b)-(f) estimates obtained using one to five frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Relative errors of the estimates (14) are give below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Statistics of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Figure 6: Difference imaging of absorption µa (top row) and scattering µ1 s (bottom row) distributions using data at different number of frequencies generated with 3 ns light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Columns from left to right: (a) Simulated target difference, and (b)-(f) estimates obtained using one to five frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Relative errors of the estimates (14) are give below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Statistics of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absolute and difference reconstructions using data at different number of frequencies were computed using data simu- lated both with 3 ns and 3 ps light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The frequencies that were used were f “ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='66, 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='33, 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='00, 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='66, 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='33 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Notice that the first frequencies utilised are close to those typically used in frequency domain DOT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absolute reconstructions using 3 ns light source are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4 and for 3 ps light source in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 5 for a different number of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, the difference images using 3 ns light source are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 6 and for 3 ps light source in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Statistics of relative errors of the reconstructions using 100 noise realisations are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 8 (b)N =1 (c) N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' =2 (d) N =3 (e) N =4 (f) N =5 (a) Target w w (g) % Error stats 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 45 40 35 30 25 0 Error 39% Error 37% Error 37% Error 38% Error 36% 5 Error40% Error37% Error36% Error36% Error30% N(b) N =1 (c) N =2 (d) N =3 (e) N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' =4 (f) N =5 (a) Target 3 w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 (g) % I Error stats 45 40 335 0 Error38% Error36% Error37% Error36% Error37% 5 4433 2 3 5 0 Error 37% Error36% Error 35% Error32% Error31% N(b) N =1 (c) N =2 (d) N =3 (e) N =4 (f) N =5 (a) Target w (g) % Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='024 stats →+ 80 丰+ 60 0 Error 72% Error68% Error65% Error69% Error 64% 4 80 3 60 5 Error 77% Error 71% Error 73% Error 73% Error74% NFigure 7: Difference imaging of absorption µa (top row) and scattering µ1 s (bottom row) distributions using data at different number of frequencies generated with 3 ps light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Columns from left to right: (a) Simulated target difference, and (b)-(f) estimates obtained using one to five frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Relative errors of the estimates (14) are give below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Statistics of estimation errors using 100 target distributions are shown as ‘boxplots’ in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Figure 8: Simulated TPSF at 1 ps interval (black solid line) on a detector located opposite to the source position in the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Sampled TPSFs at 500 ps, 1000 ps and 2000 ps temporal resolutions (blue, red and green lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The results show no apparent differences between reconstructions from data generated with nanosecond or picosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, both absolute and difference imaging reconstruct the absorbing and scattering inclusion positions with similar accuracy regardless of the number of frequencies used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' On the other hand, the contrast of the images is improved if the number of frequencies is larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, the contrast does not significantly improve if more than three frequencies are utilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This corresponds to our earlier findings showing that utilising multiple frequencies in imaging in diffuse medium improves reconstructions, but only up to few frequencies when compared to full time-domain data [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Based on these simulations, we chose to use data at three frequencies in later simulations of this work, as it provided reconstructions with adequate quality and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' If DOT measurements are performed using oscilloscopes, such as in our system described in Section 3, response time of detection is slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This can be modelled as a low-pass filter in frequency domain computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In practise, this limits the frequencies in data that can be obtained and used in reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In the experimental system used in this paper, the cut-off frequency is approximately 400 MHz, and thus the frequencies that were used were limited below this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='4 Reconstructions using data sampled at different temporal resolutions Next, the effect of low temporal sampling of data on image reconstruction was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, samples at different time-intervals were taken from the simulated time-domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The simulated data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' TPSF, on a detector located opposite to the source on the other side of the simulation domain generated by a 3 ns light pulse and the signals sampled with different temporal resolutions are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' As it can be seen, coarse sampling does not capture the original TPSF completely, that can be expected to lead artefacts in the reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, the signals with different temporal resolutions were Fourier-transformed to frequency domain where absolute and difference reconstructions were computed using three frequencies of data, f “ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='66, 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='33, 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='00 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For difference imaging, the simulated reference data was sampled using the same temporal sampling and transformed to frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absolute reconstructions from data at different temporal sampling are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 9, and the difference images from 9 X10-8 1ps xcitance O500ps 1000ps 2 2000ps 0米米 0 5 10 nanoseconds(b) N =1 (c) N =2 (d) N =3 (e) N =4 (f) N =5 (a) Target 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='024 (g) % Error stats 80 60 0 Error72% Error68% Error 66% Error 69% Error64% 4 80 3 60 0 Error 75% Error75% Error 72% Error73% Error 74% NFigure 9: Absolute imaging of absorption µa (top row) and scattering µ1 s (bottom row) distributions from 3 ns light source data sampled at different temporal discretisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Columns from left to right: (a) Simulated target, and estimates obtained by sampling the measured TPSFs at (b) 1 ps, (c) 50 ps, (d) 500 ps, (e) 1000 ps, (f) 2000 ps and (g) 3000 ps temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Relative errors of the estimates (14) are given below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Figure 10: Difference imaging of absorption δµa (top row) and scattering δµ1 s (bottom row) distributions from 3 ns light source data sampled at different temporal discretisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Columns from left to right: (a) Simulated target difference, and estimates obtained by sampling the measured TPSFs at (b) 1 ps, (c) 50 ps, (d) 500 ps, (e) 1000 ps, (f) 2000 ps and (g) 3000 ps temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Relative errors of the estimates (14) are given below each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' data at different temporal sampling are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' As it can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 9, the absolute reconstructions obtained from data with temporal sampling between 1 ps and 1000 ps look qualitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, the relative errors of the estimates are approximately the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, when temporal sampling decreases even more, the reconstructions suffer from artefacts, that are substantial at low temporal sampling (images with 2000 ps and 3000 ps temporal resolution), and the relative errors increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It can further be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 10 that difference imaging cancels out the errors caused by a low temporal sampling to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, when the sampling is in the same level as the source pulse width, the difference imaging cannot correct image artefacts either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In general, the simulations demonstrate that low temporal sampling should not affect on the reconstructions as long as the sampling is high enough to capture the temporal features of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In our experimental setup, the temporal sampling is 50 ps and that should not affect the reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 Reconstructions in the presence of light source variations Nanosecond lasers suffer from larger variations in light pulse width and shape than picosecond lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, we studied with simulations the effect of light source variations on image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The results were compared against picosecond sources with less variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For this, we measured 10 light source pulses of the laser of the experimental setup, with 50 ps sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, these 10 (a) Target (b) 1ps (c) 50ps (d) 500ps (e) 1000ps (f) 2000ps (g) 3000ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 00 0 Error=34% Error=35% Error=34% Error=36% Error=283% Error = 1884% 5 O Error = 32% Error=34% Error=35% Error=33% Error=84% Error =275%(a) Target (b) 1ps (c) 50ps (d) 500ps (e) 1000ps (f) 2000ps (g) 3000ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='02 o Error=66% Error = 67% Error=68% Error= 66% Error = 59% Error = 696% 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Error=73% Error= 76% Error =73% Error = 72% Error=76% Error=560%Figure 11: Effect of nanosecond source variations on absolute and difference reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (a) Normalised source pulses (red line) and the corresponding normalised simulated data (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (b) Simulated target, and (c) absolute absorption µa and scattering µ1 s reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (c) Simulated target difference, and (e) difference absorption δµa and scattering δµ1 s reconstructions in the presence of source variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Figure 12: Effect of picosecond source variations on absolute and difference reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (a) Simulated normalised source pulses (red line) and the corresponding normalised simulated data (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (b) Absolute absorption µa and scattering µ1 s reconstructions and (c) difference absorption δµa and scattering δµ1 s reconstructions in the presence of source variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' source signals were interpolated to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 ps duration and were used as light sources in simulations to simulate 10 TD-DOT data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These source pulses and data were averaged similarly as in data processing of the experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The reference data for difference imaging was simulated and processed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Measured source pulses are show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 11 (a) together with the corresponding simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In order to compare the reconstructions to picosecond lasers that have less variations, a proportional source pulse variations were simulated to the picosecond light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For that, 1400 source pulses were measured and their amplitude and temporal (phase) variations were calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 (arbitrary units) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='23 ns respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' These were then scaled to picosecond range and used to simulate 10 picosecond sources and simulate TD-DOT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Simulated picosecond source signals together with the corresponding simulated data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 12 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, the simulated signals were Fourier transformed to the frequency domain, and absolute and difference images were reconstructed similarly as earlier using data at three frequencies f “ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='66, 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='33, 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='00 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absolute and difference reconstructions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 11 for data simulated with nanosecond light pulses and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 12 for data simulated with picosecond light pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' As it can be seen, the source variations do not result in any additional loss of image quality, compared to absolute and difference images obtained without source pulse fluctuations in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It can also be seen that there are no significant differences between reconstructions from nanosecond and picosecond lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' That is, although nanosecond lasers have larger variations, it seems that those do not affect the accuracy of the reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 11 (a) Data (b) Target (c) Absolute (d) Target (e) Difference reconstructions reconstructions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 0 10000 Error = 36% Error=67% 5 ps 1 Error=34% Error=72%(a) Data (b) Target (c) Absolute (d) Target (e) Difference reconstructions reconstructions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='5 0 10000 Error=38% Error= 70% 4 5 ps Error=33% Error=68%Figure 13: (a) Raw measurement data from the homogeneous reference phantom showing the source pulse (red) and the measurement pulse (blue) measured using an adjacent source-detector pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (b) Mean data from 200 samples of source (red) and measurement pulses (blue) in a truncated temporal interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (c) Logarithm of amplitude (first half of the x-axis) and phase (second half of the x-axis) of the Fourier transformed measurements for all source-detector configurations from a phantom with two inclusions (red), homogeneous phantom (blue), and their difference (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 5 Experiments DOT measurements were performed using the experimental system and phantoms described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Two cylindrical phantoms with an absorbing inclusion and a scattering inclusion approximately five times the value of the background parameters were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The phantoms are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 2 together with the measurement protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The measurements were performed using eight source locations (S1-S8) and seven detector locations (D1-D7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, measurements with a homogeneous reference phantom were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The reference data was utilised in difference imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In addition, it was used to provide a computational calibration measurement for absolute imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' A raw measurement signal collected from the homogeneous reference phantom on a detector adjacent to a source is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 13 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The measurement data was averaged over 200 the samples and truncated to a temporal window of the measurement pulse width as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 13 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, the data was Fourier-transformed to the frequency domain and deconvoluted with a source by dividing the Fourier-transformed data with the corresponding Fourier-transformed source pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The Fourier transformed data at frequency f “ 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='36 MHz for all source-detector combinations is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 13 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The difference and absolute reconstructions were computed from frequency domain data at frequency f “ 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='36 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' In reconstructions, a 2D computation domain was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The domain was discretised using 1369 nodes and 2622 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absolute reconstructions were computed by minimising (9) and the difference reconstructions were computed by minimising (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The minimisation problems were solved using Gauss-Newton method similarly as for simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For absolute imaging, the measurements with inclusions were precalibrated using the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Calibration coefficients for individual source-detector pairs were computed by taking the difference between the experimental refer- ence measurements and simulated reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, this difference was subtracted from the experimental measurements with inclusions, and the subtracted measurements were considered as calibrated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Reconstructed absolute and difference images for the two phantoms are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' As it can be seen, the location of absorption and scattering inclusions can be distinguished both using absolute and difference imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Also the difference in inclusion positions between the two phantoms is clearly visible both in absolute and difference imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The absorption images show some cross-talk from the scattering inclusion, that is especially evident in the difference images, but the magnitude of the cross-talk is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The scattering images, on the other hand, have more artefacts both in absolute and difference imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The difference imaging shows less artifacts than absorption images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' That is typical in DOT since difference imaging compensates both measurement and modelling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Overall, all reconstructions can be regarded as good quality DOT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, the estimated inclusion values do not reach the correct absolute values of the inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This can be due to, for example, relatively low number of measurement positions that are all located on a single plane of the 3D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 6 Discussion and conclusions In this work, feasibility of utilising nanosecond sources in TD-DOT was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Different aspects and possible limitations of TD-DOT systems with nanosecond sources were studied with simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' First, frequency content of time-domain DOT data simulated with nanosecond light sources was studied and compared against picosecond source systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The simulations verified the previous findings [27] that information of diffuse medium is 12 (a)Rawdata 【b)Truncated(mean)data ()Fouriertransformed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='2 (0) 0 50 100 150 0 10 0 50 100 Time (ns) Time (ns)Figure 14: (a) Reconstructed absolute absorption µa (first row) and scattering µ1 s (second row) images for the two phantoms (first and second column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' (b) Reconstructed difference absorption δµa (first row) and scattering δµ1 s (second row) images for the two phantoms (third and fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' on low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This enables image reconstruction using Fourier-transformed time-domain data using few frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, it enables usage of measurement electronics with slower response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Second, effect of temporal sampling was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The simulations showed that temporal sampling needs to be high enough to capture the TPSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' This sampling can be achieved with standard digital oscilloscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Third, nanosecond lasers can suffer from light pulse variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It was shown that, although these variations are large, they do not affect significantly on image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Overall, the simulations demonstrated the capability of nanosecond sources to be utilised in TD-DOT in diffuse medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Then, a prototype TD-DOT experimental system utilising a high-energy nanosecond source was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The system consisted of a nanosecond Nd:YAG laser combined with optical parametric oscillator for light input, optical fibres and collimators for guiding the light, and avalanche photodetector and high-bandwidth oscilloscope for measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The system is relatively robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' For example, detector is temperature-compensated, it does not require cooling and can be operated in normal room lighting without saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The pros of the system include broad wavelength tuning range for multiwavelength imaging as well as high energy per wavelength and pulse, that would enable directing the light to multiple fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Further, relative impacts of pulse temporal spreading in long optical fibres, inaccuracies on fibre lengths, or changes in environmental conditions such as temperature and vibrations are smaller for nanosecond pulses and could provide robustness and stability for the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Furthermore, a nanosecond laser and a established digitizer based signal detection enable compatibility of the system with other techniques such as ultrasound imaging (signal waveform detection) and photoacoustic tomography (signal waveform detection and laser source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The cons of the system include poor laser pulse stability, that was not an issue in the experiments of the study but can be significant in some other applications, and the low pulse repetition frequency of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Also the dynamic range and sensitivity of the system may have limitations in larger targets, which would require more research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The DOT system was used in both absolute and difference imaging of two phantoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' It was shown that both absorbing and scattering objects could be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The locations of the inclusions were found, and the cross-talk between the absorbing and scattering targets was low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The reconstructions could be improved, for example, by adding more measurement layers to the phantom and extending modelling to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The simulations and experiments of this work were the first study demonstrating usage of nanosecond sources in TD- DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The nanosecond system can be utilised in diffuse medium, that is in highly scattering medium when the imaged target size is larger than multiple free scattering lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' However, its performance, for example, in dilute medium where photons travel faster from source to detector or with measurement setups with short source-detector distances were not studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Overall, developments of DOT systems are guided by potential applications and their different requirements on, for example, imaging depth, speed, invasiveness, scalability and multimodality [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Therefore, further work is needed for the development of the proposed setup, and finding the applications where it could be seen as most beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The system could be further developed by building source energy specific light detection utilising, for example, neutral density filters or laser beam attenuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' Sensitivity of the system could be increased using large diameter detection fibre bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' 13 (a) Absolute reconstructions (b) Absolute reconstructions (c) Difference reconstructions(d) Difference reconstructions phantom 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0084 phantom 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0080 phantom 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0012 phantom 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='0057 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content='90Acknowledgments This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101001417- QUANTOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfkASI/content/2301.03269v1.pdf'} +page_content=' The work has been supported by the Academy of Finland (projects 314411, 336799 Centre of Excellence in 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0000000000000000000000000000000000000000..85bbce53fe7c36828010e171dea22a045f5951b6 --- /dev/null +++ b/vdAzT4oBgHgl3EQf7P4z/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:599e2666c63f4dde905fe4a9a9f5a712f3bd6b38e740fc4665f752bb69d6201a +size 4456493 diff --git a/w9FAT4oBgHgl3EQfjh0a/content/tmp_files/2301.08605v1.pdf.txt b/w9FAT4oBgHgl3EQfjh0a/content/tmp_files/2301.08605v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e02e6a3ed656e9e07101a5c10f4861e9f471d2e --- /dev/null +++ b/w9FAT4oBgHgl3EQfjh0a/content/tmp_files/2301.08605v1.pdf.txt @@ -0,0 +1,692 @@ +SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 +1 +A Deep Learning Approach for SAR Tomographic +Imaging of Forested Areas +Zo´e Berenger, Lo¨ıc Denis, Senior Member, IEEE, Florence Tupin, Senior Member, IEEE, +Laurent Ferro-Famil, Member, IEEE, and Yue Huang +Abstract—Synthetic aperture radar tomographic imaging re- +constructs the three-dimensional reflectivity of a scene from a set +of coherent acquisitions performed in an interferometric confi- +guration. In forest areas, a large number of elements backscatter +the radar signal within each resolution cell. To reconstruct the +vertical reflectivity profile, state-of-the-art techniques perform +a regularized inversion implemented in the form of iterative +minimization algorithms. We show that light-weight neural net- +works can be trained to perform the tomographic inversion with +a single feed-forward pass, leading to fast reconstructions that +could better scale to the amount of data provided by the future +BIOMASS mission. We train our encoder-decoder network using +simulated data and validate our technique on real L-band and +P-band data. +Index Terms—SAR tomography (TomoSAR), deep learning, +forests, inverse problems +I. INTRODUCTION +S +YNTHETIC aperture radar (SAR) tomography (Tomo- +SAR) uses a 2D aperture to perform 3D imaging. In +the case of a narrow-band radar waveform and under the +widely adopted Born approximation at order 1 [1], the imaging +process simplifies to the 1D spectral analysis of a set of +co-registered 2D SAR images [2]. It aims at reconstructing, +for each 2D location, reflectivity profiles in the direction +orthogonal to the radar line-of-sight. As illustrated in [3], +parametric spectral estimation approaches, which have been +widely used for the characterization of urban areas, such as +high-resolution techniques [3] or Compressive Sensing (CS)- +inspired regularized least squares minimization [4], [5], fail +to adequately reconstruct the response of forested environ- +ments. They indeed estimate a small set of discrete point-like +scattering sources, instead of a continuous function, known to +represent well the reflectivity of such volumetric media [6]. +Among the wide range of existing non-parametric spectral +estimation techniques [7], the beamformer, i.e. the discrete +Fourier transform, and Capon’s filter, also called the adaptive +beamformer, are the most widely used to perform TomoSAR +This project has been funded by the Futur & Ruptures PhD program of the +Fondation Mines-Telecom, and partially funded by ASTRAL project (ANR- +21-ASTR-0011). +Z. Berenger and F. Tupin are with LTCI, T´el´ecom Paris, Institut polytech- +nique de Paris, Paris, France (e-mail: name.surname@telecom-paris.fr). +L. Denis is with the Laboratoire Hubert Curien, UMR 5516, CNRS, Institut +d’Optique Graduate School, Univ. Lyon, UJM-Saint-´Etienne, Saint-´Etienne +42023, France (e-mail: loic.denis@univ-st-etienne.fr). +L. Ferro-Famil is with ISAE-SUPAERO and CESBIO, University of +Toulouse, Toulouse, France (e-mail: laurent.ferro-famil@isae-supaero.fr). +Y. Huang is with CESBIO, University of Toulouse, Toulouse, France (e- +mail: yhuang228@gmail.com). +focusing over forests. The beamformer has a coarse resolution +and creates sidelobes, whereas Capon’s sidelobe reduction +capability comes at the cost of radiometric accuracy [7]. A +parametric solution based on the use of a sparsifying basis able +to approximate a continuous function using a small set of co- +efficients was proposed in [8]. This approach, named wavelet- +based CS, used a CS-inspired optimization to determine a +reflectivity profile constructed using an orthogonal wavelet +matrix and a regularized number of wavelet coefficients. +Another approach, using a small number of parametric basis +functions, was proposed in [9]. The regularized inversion of a +linear model of the covariance matrix leads to much more ac- +curate reconstructions, nevertheless the latter estimators, which +are themselves non-linear, require costly iterative minimization +algorithms that impede their application to large-scale datasets. +In many imaging domains, deep learning has made it +possible to reduce computation time while maintaining high- +resolution results. This potential has been harnessed for SAR +tomography in [10], where the authors unroll an Iterative +Shrinkage Thresholding Algorithm (ISTA) to solve the L2−L1 +norm minimization problem posed by CS in urban areas. This +approach has been improved in [11], [12], yet most deep +learning techniques for SAR tomography over forests focus +on ground and canopy height estimation using LiDAR data as +a reference [13]. +This paper presents a supervised deep learning method +for tomographic SAR reconstruction in forested areas. Its +objective is to recover the reflectivity profile from the coarse +profile obtained by the beamforming algorithm. We first +use a physics-inspired generation model and interferometric +baselines matching our SAR dataset to simulate reflectivity +distributions and associated measurements. We then train a +network with a light-weight architecture to learn a low- +dimensional latent representation of these simulated profiles +and to recover the original profiles free from beamforming +artifacts. Finally, the neural network is evaluated on real +beamforming profiles at L-band and P-band and compared +to several methods, showing promising performances both in +terms of reconstruction quality and computation time. +II. METHODOLOGY +A. Problem formulation +The tomographic signal measured over N SAR acquisitions, +y ∈ CN, may be formulated as the sum of the Ns contributions +arXiv:2301.08605v1 [eess.IV] 20 Jan 2023 + +SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 +2 +originating from the considered 2D resolution cells as: +y = +Ns +� +k=1 +sk a(zk) + ϵ = As + ϵ +(1) +where s = [s1, . . . , sNs]T contains the complex reflection +coefficient of the observed scatterers, a(z) = [1, . . . , ejk(N) +z +z]T +is a steering vector and models the interferometric phase with +k(n) +z +z corresponding to the phase seen on the n-th image for +a scatterer located at height z, A = [a(z1), . . . , a(zNs)] is the +sensing matrix, while ϵ stands for the system noise, distributed +according to a circular complex Gaussian distribution of +covariance σ2 +ϵ IN. +As mentioned earlier, over forested areas, the vertical den- +sity of reflectivity s is well modelled by a speckle-affected +continuous function, and hence may be represented in (1) by +a large number Ns ≫ N of uncorrelated source reflectivities +distributed over a vector z ∈ RNz of Nz discrete heights. +The covariance matrix of s is diagonal and is given by +Σss = E[ssH] = diag(p), with ·H the Hermitian transpose +operator. The measured signal covariance matrix can then be +expressed as: +Σ = E[yyH] = Adiag(p)AH + σ2 +ϵ IN +(2) +In practice, this quantity is estimated using L independent +realizations of the measured vector, {yl}L +l=1, sampled in the +neighborhood of a 2D location. The corresponding sample +covariance matrix is given by �Σ = +1 +L +�L +l=1 ylyH +l . The +objective of forest TomoSAR imaging is the estimation, or +at least the characterization, of p ∈ RNz ++ +from �Σ. The sample +correlation matrix, �R = diag(q)�Σdiag(q), with qi = 1/ +� +�Σii, +represents a version of �Σ, in which the intensity of each +image is scaled to 1. Such a representation, independent of +the observed absolute reflectivity, may also be used during +specific steps of the 3D imaging process. +B. From fixed dictionaries to learned representations: strate- +gies for vertical profile reconstruction +As shown in [3], [8], forest reflectivity profiles p can +typically be approximated as a linear combination of a few +basis functions: +p ≈ Ψα +(3) +where Ψ ∈ RNz×Nα is the dictionary whose columns are the +basis functions and α ∈ RNα is the vector of weights. For +an adequately chosen basis, only a few functions at a time +are necessary to approximate a given profile. The vector α is +then sparse: most coefficients in α are zero and the number +of non-zeros ∥α∥0 is small, as is the L1 norm ∥α∥1 which is +often used as a proxy to measure sparsity. The reflectivity +profile �p(CS) = Ψ�α can be estimated using the following +minimization problem [8]: +�p(CS) = arg min +p≥0 +∥Adiag(p)AH − �Σ∥2 +F + λ∥Ψ†p∥1 . +(4) +where λ is a hyper-parameter responsible for balancing the +weight of the sparsity constraint, with respect to the data- +fidelity term defined by the Frobenius norm. The pseudo +inverse matrix, Ψ†, in the L1 norm term verifies ΨΨ†p = α, +and may be replaced with ΨH in the case of a unitary orthog- +onal basis. Solving this problem typically requires several tens +or hundreds of iterations of a minimization algorithm and the +proper tuning of the hyper-parameter λ. +Rather than using a linear model for the profiles p, non- +linear models such as the generative model discussed in +paragraph II-D can be used (e.g., Gaussian mixture models, or +exponential profiles [9]). However, inverting such models can +be difficult due to identifiability issues and require additional +constraints such as order constraints as recently proposed in +a different modality of remote sensing for the estimation of +phenological parameters from NDVI time series [14]. Our +early experiments have shown that it was preferable to jointly +learn a non-linear model for p = m(α) and an estimator �α +by training a deep neural network with an encoder-decoder +architecture. The code α in the latent space learnt by the en- +coder is then turned into a profile by the decoder. The use of a +simple feed-forward pass of a light-weight neural architecture +leads to an extremely efficient tomographic inversion method, +as described in the following paragraphs. +C. Proposed method +The different steps of our method are presented in Fig. 1. +Starting from a ground truth profile created with a defined +generative model, SAR measurements y can be simulated +using the steering matrix A of a specific geometric confi- +guration. To ease the network training, we propose to provide +as input to the network the beamforming profile computed +from a multi-look correlation matrix �R obtained from L +measurements y1 to yL. In this way, the network input and +output are in the same space, i.e. both positive-valued vectors, +easier to manipulate than complex-valued vectors. Besides, +the beamforming profile depends less on the interferometric +baselines than the correlation matrix �R. It includes the physics +of acquisition through the steering matrix. +The correlation matrix has been used rather than the co- +variance matrix �Σ to make the method invariant to differing +intensity values. The final reconstruction will therefore be +�p = Tr(�Σ/N) ˜p, with ˜p the output of the network, to recover +the actual intensity value of the profile. +We have chosen to use a simple network architecture with +an encoder-decoder going through a latent space of reduced +dimension. The choice of this dimension is made to represent +the number of parameters describing the profile. The loss +used is a quadratic loss between the ground-truth simulated +profile and the output profile, a standard choice for regression +problems. +D. Generative model +The distribution of reflectivity in a forest area cell is mainly +composed of two peaks, one for the ground and one for the +canopy, as validated by the latest tomographic reconstruction +methods [8], [9]. These peaks can be represented by many + +SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 +3 +Fig. 1. Pipeline of the data simulation, training and testing process of the proposed method. +basis functions, of which the simplest case is that of two +Gaussians. The proposed approach therefore aims at learning +a latent representation of a forest reflectivity profile from +simulated profiles composed of two Gaussians. The parameters +of these Gaussians are adapted to the type of forests observed +in our SAR datasets for each training, i.e. boreal forests in +L band and tropical forests in P band. The ranges of these +parameters are chosen to be large, to cover all possible profiles +met in practice. The influence of the choice of these parameters +is discussed in section IV. +III. EXPERIMENTS +A. Simulated data +10000 simulated profiles sampled on Nz uniformly spaced +heights were generated with a mixture of two Gaussians +with various parameters. The corresponding covariance ma- +trices Σ = Adiag(p)AH were built with a steering ma- +trix A randomly selected among steering matrices of an +actual tomographic dataset. Circular complex Gaussian sam- +ples {yl}l=1..L were then drawn according to these covari- +ance matrices (by multiplying a white speckle noise wl by +Adiag(√p)). Sample correlation matrices �R were computed +from each set of L samples yl. The beamforming profiles were +then reconstructed: +˜p(B) +i += aH(zi) �Ra(zi) +N 2 +(5) +where a is a steering column vector of the matrix A. These +beamforming profiles were then used as input by the neural +network. +The trained encoder consists of 4 linear layers with a +decreasing number of neurons (starting with Nz = 512 and +encoding the input data in a latent space of size 5) and a +symmetric decoder of 4 linear layers. The depth of the network +is voluntarily kept small and has been empirically validated, +as discussed in section III-A2. Each layer is unbiased and has +a leaky ReLU activation function. Training was performed on +the dataset consisting of the simulated beamforming profiles, +divided into training (75%) and validation (25%) sets, in mini- +batches of size 32 with an Adam optimizer and a learning rate +of 10−3 for 200 epochs. +In the following subsections, the results presented were +computed using the steering matrices of the BioSAR-2 cam- +paign, more thoroughly presented in section III-B. +1) Analysis of the reconstructed profiles: Fig. 2 shows, for +two different simulated profiles, the reconstructions obtained +with the beamforming algorithm, Capon filter, wavelet-based +CS and our method for 100 speckle realizations in the measure- +ment simulation. As we are interested in comparing the shapes +of the profiles reconstructed by each algorithm and since +they do not compute the same physical quantity, the methods +corresponding to the output power of a filter have been plotted +above and the reconstructed and reference reflectivity profiles +below. The left profile composed of two large Gaussians favors +the spectral estimation methods and shows that the wavelet- +based CS fails to reconstruct large lobes, while the profile on +the right simulates very narrow responses, more suitable for +the latter but only roughly reconstructed by Beamforming for +the level of noise considered (L = 100). As for the Capon +filter, it performs well with low thermal noise, but is very +affected when the noise increases, even with suitable diagonal +loading. The neural network produces profiles with a width +that follows more closely the ground-truth profiles in each +case. +(a) Beamforming (blue) +Capon (green) +(b) Beamforming (blue) +Capon (green) +(c) Ground-truth (black) +Proposed method (red) +Wavelet-based CS (magenta) +(d) Ground-truth (black) +Proposed method (red) +Wavelet-based CS (magenta) +Fig. 2. Reconstruction of a profile consisting of wide (left) and narrow (right) +Gaussians with Beamforming, Capon, Wavelet-based CS and our method. +For each approach, the average profile over 100 measurements with various +speckle realizations and its interquartile range have been plotted, with the +reference profile shown in black. +2) Architecture choice: Different latent space sizes were +tested to support the intuition of keeping a size corresponding + +le-1 +4 +0 +-10 +0 +10 +20 +301 +0 +-10 +0 +10 +20 +301e-2 +2 +0 +10 +0 +10 +20 +30le-1 +1 +0 +10 +0 +10 +20 +30SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 +4 +TABLE I +MEAN AND STANDARD DEVIATION OF THE MEAN SQUARED ERROR (MSE) VALUES AFTER 20 TRAININGS DURING 200 EPOCHS FOR EACH DIFFERENT +LATENT SPACE SIZES, COMPARED TO A NORMALIZED ERROR BETWEEN THE INPUT BEAMFORMING AND REFERENCE PROFILE. +Latent space size +3 +4 +5 +6 +8 +10 +15 +20 +MSE (×10−1) +5.12 ± 0.10 +4.74 ± 0.12 +4.65 ± 0.15 +4.68 ± 0.19 +4.64 ± 0.16 +4.68 ± 0.18 +4.67 ± 0.15 +4.68 ± 0.13 +to the number of parameters needed to simulate a mixture of +two Gaussians. The number of looks L was set to 100 for this +study. The reconstruction errors, compared to a normalized +error between the input beamforming �p(B) and reference profile +p, ∥�p(B) ⟨�p(B) , p⟩ +⟨�p(B) , �p(B)⟩ −p∥2 +2, where ⟨· , ·⟩ is the L2 scalar product, +are presented in Table I. They indicate that a latent space of +size smaller than 5 significantly increases the reconstruction +error, while a larger latent space brings only marginal improve- +ments. +B. Boreal forest at L band +Testing was first performed on a tomographic stack of 6 +airborne L-band SAR images of a boreal forest in northern +Sweden, acquired during the BioSAR-2 campaign led by the +DLR in 2008 [15], with a vertical resolution varying from +6 m in near range to 25 m in far range. A local window of +around 60 looks is used to compute the covariance matrix used +by the beamforming algorithm to compute the tomograms. +Fig. 3 shows an example of a tomogram in HH polarization +reconstructed with the proposed method, after compensation +of the topography. It is compared to the results obtained +with beamforming, Capon and the wavelet-based CS method +developed in [8], showing that the trained network does indeed +improve the resolution of the tomogram compared to spectral +estimation methods, while maintaining a representative volume +of the tree crown. This is not the case with the wavelet-based +CS, which reconstructs narrow peaks and can therefore predict +several peaks when this volume is large, depending on the +choice of the regularization parameter value. +The orders of magnitude of the computational time required +to compute these tomograms are given in Table II (all compu- +tations are done on one CPU, even the network training). The +wavelet-based CS approach was computed using the optimi- +sation library CVX and the default solver SDPT3 [16]. These +statistics highlight one of the major advantages of using deep +learning to reconstruct forest reflectivity profiles, which is the +gain in time of this method compared to classical optimization +algorithms. This feature will be even more important when +scaling up and applying tomographic reconstruction on an +entire SAR image. +C. Tropical forest at P band +Other tests were performed on airborne tomographic SAR +data acquired at P band by the ONERA over the test site of +Paracou in French Guiana, during the TropiSAR campaign in +2009 [17], also comprising 6 tracks, with a vertical resolution +of around 15 m. The correlation matrix used to compare the +different methods is calculated with 56 looks. A sample of the +results and comparisons are presented in Fig. 4. The proposed +(a) Beamforming +(b) Capon +(c) Wavelet-based CS +(d) Proposed method (deep learning) +Fig. 3. Tomogram for a specific azimuth value in a boreal forest at L band +derived with the method: (a) Beamforming; (b) Capon; (c) Wavelet-based CS; +(d) Predicted by the proposed neural network. +TABLE II +ORDER OF MAGNITUDE OF THE COMPUTATION TIME FOR THE +RECONSTRUCTION OF A TOMOGRAM OF SIZE 1.4 KM × 1.6 M ON A CPU. +Method +Computation time (s) +Beamforming +2 +Capon +3 +Wavelet-based CS (CVX) +1500 +Proposed method +� +Training +200 +Inference +3 +approach here seems even more adapted to tropical than boreal +forests, providing high resolution reconstructed profiles with +fewer information loss than with the wavelet-based CS. +IV. DISCUSSION +During training, the network is fed with beamforming pro- +files computed from samples generated according to a model +of covariance matrix. This model, defined in equation (2), +neglects several phenomena such as the integration of back- +scattered complex amplitudes produced by scatterers located +in the neighborhood according to the SAR impulse response, +or the temporal decorrelation between acquisitions. It could +be refined, but our results on real data already show a good + +30 +height z [m] +20 +-10 +10 +-20 +0 +30 +10. +40 +200 +400 +600 +800 +range [m]30 +w] +20 +10 +N +height 2 +10 +20 +0 +30 +10 +40 +200 +400 +600 +800 +range [bin]30 +0 +[w] +20 +-10 +height z [ +10 +-20 +0 +30 +-10 +40 +200 +400 +600 +800 +range [m]30 +0 +height z [m] +20 +-10 +10 +-20 +0 +-30 +-10 +40 +0 +200 +400 +600 +800 +range [m]SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 +5 +(a) Beamforming +(b) Capon +(c) Wavelet-based CS +(d) Proposed method (deep learning) +Fig. 4. Tomogram for a specific azimuth value in a tropical forest at P band +derived with the method: (a) Beamforming; (b) Capon; (c) Wavelet-based CS; +(d) Predicted by the proposed neural network. +generalization ability of the network trained with this simple +covariance model. +For the training, we used the steering matrix of a specific +campaign to simulate measurements and beamforming profiles +that match at best the actual SAR data considered at test time. +The network is thus trained for a given geometric configuration +and the efficiency of the network may be affected when it +differs, in which case a retraining would be required. However, +this step is relatively inexpensive in terms of computational +time (a few minutes), and tomographic data acquisition con- +figurations in a given spectral band vary only slightly due to +hardware constraints and the possibility of ambiguities. +In our experiments, the network was trained to reconstruct +profiles composed of two Gaussians. It cannot therefore per- +fectly recover a profile with a single or an extra peak, even +with input profiles composed of three very distinct Gaussians. +The use of a refined simulation model could further improve +the reconstruction quality, which encourages current research +on a model representative of different forest types [9]. +V. CONCLUSION +In this study, we explored the possibility of using deep +learning approaches to improve the performance of the to- +mographic inversion task on forests. Tests on real data show +promising results, both in terms of quality and in terms of +the computation time needed to reconstruct a large image, +with an overhead for the feedforward pass negligible com- +pared to the computation of the beamforming profile and +an acceleration by several orders of magnitude with respect +to iterative regularized inversion algorithms. This will be +essential for the systematic processing of future data from the +ESA BIOMASS [18] mission. It confirms the strong potential +of the application of deep learning in this field to super-resolve +existing reconstructions. +It could also be interesting to consider directly using the +measured intensities and acquisition geometry as input to the +neural network, to avoid the need to train a different network +for each tomographic configuration. +REFERENCES +[1] L. Ferro-Famil, Y. Huang, and E. Pottier, “Principles and applications +of polarimetric SAR tomography for the characterization of complex +environments,” International Association of Geodesy Symposia. F. Sanso +Ed.., Springer-Verlag, vol. 142, no. 1-13, pp. 243–255, 2016. +[2] F. Gini and F. Lombardini, “Multibaseline cross-track SAR interferom- +etry: a signal processing perspective,” IEEE Aerospace and Electronic +Systems Magazine, vol. 20, no. 8, pp. 71–93, 2005. +[3] Y. Huang, J. L´evy-Vehel, L. Ferro-Famil, and A. Reigber, “Three- +dimensional imaging of objects concealed below a forest canopy using +SAR tomography at L-band and wavelet-based sparse estimation,” IEEE +Geoscience and Remote Sensing Letters, vol. 14, no. 9, pp. 1454–1458, +Sep. 2017. +[4] A. Budillon, A. Evangelista, and G. Schirinzi, “Three-dimensional SAR +focusing from multipass signals using compressive sampling,” IEEE +Transactions on Geoscience and Remote Sensing, vol. 49, no. 1, pp. +488–499, 2011. +[5] C. Rambour, A. Budillon, A. C. Johnsy, L. Denis, F. Tupin, and G. Schir- +inzi, “From interferometric to tomographic SAR: A review of synthetic +aperture radar tomography-processing techniques for scatterer unmixing +in urban areas,” IEEE Geoscience and Remote Sensing Magazine, vol. 8, +no. 2, pp. 6–29, 2020. +[6] H. Aghababaee, G. Ferraioli, L. Ferro-Famil, Y. Huang, M. Mari- +otti D’Alessandro, V. Pascazio, G. Schirinzi, and S. Tebaldini, “Forest +SAR tomography: Principles and applications,” IEEE Geoscience and +Remote Sensing Magazine, vol. 8, no. 2, pp. 30–45, Feb. 2020. +[7] P. Stoica and R. L. Moses, Spectral analysis of signals. +Upper Saddle +River, NJ: Prentice Hall, 2005. +[8] E. Aguilera, M. Nannini, and A. Reigber, “Wavelet-based compressed +sensing for SAR tomography of forested areas,” IEEE Transactions on +Geoscience and Remote Sensing, vol. 51, no. 12, pp. 5283–5295, 2013. +[9] L. Ferro-Famil, Y. Huang, and N. Ge, “Estimation of the vertical +structure of a tropical forest using basis functions and parametric SAR +tomography,” in IGARSS, 2022. +[10] K. Qian, Y. Wang, Y. Shi, and X. X. Zhu, “Super-resolving SAR +tomography using deep learning,” in IGARSS, 2021, pp. 4810–4813. +[11] ——, “γ-Net: superresolving SAR tomographic inversion via deep learn- +ing,” IEEE Transactions on Geoscience and Remote Sensing, vol. 60, +pp. 1–16, 2022. +[12] Y. Hu, X. Zhang, S. Wei, Y. Ren, N. Wang, and J. Shi, “Adatomo- +Net: a novel deep learning approach for SAR tomography imaging and +autofocusing,” in IGARSS, 2022. +[13] W. Yang, A. Budillon, G. Ferraioli, V. Pascazio, G. Schirinzi, and +S. Vitale, “A deep learning solution for height reconstruction in SAR +tomography,” in IGARSS, 2022. +[14] Y. Z´erah, S. Valero, and J. Inglada, “Physics-guided interpretable prob- +abilistic representation learning for high resolution image time series,” +IEEE Transactions on Geoscience and Remote Sensing, 2022. +[15] I. Hajnsek, R. Scheiber, M. Keller, R. Horn, S. Lee, L. Ulander, +A. Gustavsson, G. Sandberg, T. Toan, S. Tebaldini, A. Guarnier, and +F. Rocca, BIOSAR 2008 technical assistance for the development of +airborne SAR and geophysical measurements during the BioSAR 2008 +experiment, final report, 2009. +[16] M. Grant and B. SP, “CVX: MATLAB software for disciplined convex +programming,” 2014. [Online]. Available: http://cvxr.com/cvx +[17] P. Dubois-Fernandez, T. L. Toan, S. Daniel, H. M. Oriot, J. Chave, +L. Blanc, L. Villard, M. Davidson, and M. Petit, “The TropiSAR air- +borne campaign in French Guiana: objectives, description, and observed +temporal behavior of the backscatter signal,” IEEE Transactions on +Geoscience and Remote Sensing, vol. 50, pp. 3228–3241, 2012. +[18] K. Fletcher and H. Rider, “Report for mission selection: an earth +explorer to observe atmospheric composition. BIOMASS.” ser. ESA +SP-1324/1 (3 volume series), European Space Agency, Noordwijk, The +Netherlands, 5 2012. + +60 +0 +height z [m] +40 +-10 +20 +-20 +0 +30 +20 +40 +0 +250 +500 +750 +1000 +range [m]60 +0 +height z [m] +40 +-10 +20 +-20 +0 +30 +20 +40 +0 +250 +500 +750 +1000 +range [bin]60 +0 +height z [m] +40 +-10 +20 +-20 +0 +-30 +-20 +40 +0 +250 +500 +750 +1000 +range [m]60 +0 +height z [m] +40 +-10 +-20 +-30 +-20 +40 +0 +250 +500 +750 +1000 +range [m] \ No newline at end of file diff --git a/w9FAT4oBgHgl3EQfjh0a/content/tmp_files/load_file.txt b/w9FAT4oBgHgl3EQfjh0a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db52b9b9984b6f4e525b48e7df41a0bbcdfe3917 --- /dev/null +++ b/w9FAT4oBgHgl3EQfjh0a/content/tmp_files/load_file.txt @@ -0,0 +1,343 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf,len=342 +page_content='SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 1 A Deep Learning Approach for SAR Tomographic Imaging of Forested Areas Zo´e Berenger, Lo¨ıc Denis, Senior Member, IEEE, Florence Tupin, Senior Member, IEEE, Laurent Ferro-Famil, Member, IEEE, and Yue Huang Abstract—Synthetic aperture radar tomographic imaging re- constructs the three-dimensional reflectivity of a scene from a set of coherent acquisitions performed in an interferometric confi- guration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' In forest areas, a large number of elements backscatter the radar signal within each resolution cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' To reconstruct the vertical reflectivity profile, state-of-the-art techniques perform a regularized inversion implemented in the form of iterative minimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' We show that light-weight neural net- works can be trained to perform the tomographic inversion with a single feed-forward pass, leading to fast reconstructions that could better scale to the amount of data provided by the future BIOMASS mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' We train our encoder-decoder network using simulated data and validate our technique on real L-band and P-band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Index Terms—SAR tomography (TomoSAR), deep learning, forests, inverse problems I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' INTRODUCTION S YNTHETIC aperture radar (SAR) tomography (Tomo- SAR) uses a 2D aperture to perform 3D imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' In the case of a narrow-band radar waveform and under the widely adopted Born approximation at order 1 [1], the imaging process simplifies to the 1D spectral analysis of a set of co-registered 2D SAR images [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It aims at reconstructing, for each 2D location, reflectivity profiles in the direction orthogonal to the radar line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' As illustrated in [3], parametric spectral estimation approaches, which have been widely used for the characterization of urban areas, such as high-resolution techniques [3] or Compressive Sensing (CS)- inspired regularized least squares minimization [4], [5], fail to adequately reconstruct the response of forested environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' They indeed estimate a small set of discrete point-like scattering sources, instead of a continuous function, known to represent well the reflectivity of such volumetric media [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Among the wide range of existing non-parametric spectral estimation techniques [7], the beamformer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' the discrete Fourier transform, and Capon’s filter, also called the adaptive beamformer, are the most widely used to perform TomoSAR This project has been funded by the Futur & Ruptures PhD program of the Fondation Mines-Telecom, and partially funded by ASTRAL project (ANR- 21-ASTR-0011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Berenger and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Tupin are with LTCI, T´el´ecom Paris, Institut polytech- nique de Paris, Paris, France (e-mail: name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='surname@telecom-paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Denis is with the Laboratoire Hubert Curien, UMR 5516, CNRS, Institut d’Optique Graduate School, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Lyon, UJM-Saint-´Etienne, Saint-´Etienne 42023, France (e-mail: loic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='denis@univ-st-etienne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Ferro-Famil is with ISAE-SUPAERO and CESBIO, University of Toulouse, Toulouse, France (e-mail: laurent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='ferro-famil@isae-supaero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Huang is with CESBIO, University of Toulouse, Toulouse, France (e- mail: yhuang228@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' focusing over forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The beamformer has a coarse resolution and creates sidelobes, whereas Capon’s sidelobe reduction capability comes at the cost of radiometric accuracy [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' A parametric solution based on the use of a sparsifying basis able to approximate a continuous function using a small set of co- efficients was proposed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This approach, named wavelet- based CS, used a CS-inspired optimization to determine a reflectivity profile constructed using an orthogonal wavelet matrix and a regularized number of wavelet coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Another approach, using a small number of parametric basis functions, was proposed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The regularized inversion of a linear model of the covariance matrix leads to much more ac- curate reconstructions, nevertheless the latter estimators, which are themselves non-linear, require costly iterative minimization algorithms that impede their application to large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' In many imaging domains, deep learning has made it possible to reduce computation time while maintaining high- resolution results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This potential has been harnessed for SAR tomography in [10], where the authors unroll an Iterative Shrinkage Thresholding Algorithm (ISTA) to solve the L2−L1 norm minimization problem posed by CS in urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This approach has been improved in [11], [12], yet most deep learning techniques for SAR tomography over forests focus on ground and canopy height estimation using LiDAR data as a reference [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This paper presents a supervised deep learning method for tomographic SAR reconstruction in forested areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Its objective is to recover the reflectivity profile from the coarse profile obtained by the beamforming algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' We first use a physics-inspired generation model and interferometric baselines matching our SAR dataset to simulate reflectivity distributions and associated measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' We then train a network with a light-weight architecture to learn a low- dimensional latent representation of these simulated profiles and to recover the original profiles free from beamforming artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Finally, the neural network is evaluated on real beamforming profiles at L-band and P-band and compared to several methods, showing promising performances both in terms of reconstruction quality and computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Problem formulation The tomographic signal measured over N SAR acquisitions, y ∈ CN, may be formulated as the sum of the Ns contributions arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='08605v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='IV] 20 Jan 2023 SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 2 originating from the considered 2D resolution cells as: y = Ns � k=1 sk a(zk) + ϵ = As + ϵ (1) where s = [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' , sNs]T contains the complex reflection coefficient of the observed scatterers, a(z) = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' , ejk(N) z z]T is a steering vector and models the interferometric phase with k(n) z z corresponding to the phase seen on the n-th image for a scatterer located at height z, A = [a(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' , a(zNs)] is the sensing matrix, while ϵ stands for the system noise, distributed according to a circular complex Gaussian distribution of covariance σ2 ϵ IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' As mentioned earlier, over forested areas, the vertical den- sity of reflectivity s is well modelled by a speckle-affected continuous function, and hence may be represented in (1) by a large number Ns ≫ N of uncorrelated source reflectivities distributed over a vector z ∈ RNz of Nz discrete heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The covariance matrix of s is diagonal and is given by Σss = E[ssH] = diag(p), with ·H the Hermitian transpose operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The measured signal covariance matrix can then be expressed as: Σ = E[yyH] = Adiag(p)AH + σ2 ϵ IN (2) In practice, this quantity is estimated using L independent realizations of the measured vector, {yl}L l=1, sampled in the neighborhood of a 2D location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The corresponding sample covariance matrix is given by �Σ = 1 L �L l=1 ylyH l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The objective of forest TomoSAR imaging is the estimation, or at least the characterization, of p ∈ RNz + from �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The sample correlation matrix, �R = diag(q)�Σdiag(q), with qi = 1/ � �Σii, represents a version of �Σ, in which the intensity of each image is scaled to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Such a representation, independent of the observed absolute reflectivity, may also be used during specific steps of the 3D imaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' From fixed dictionaries to learned representations: strate- gies for vertical profile reconstruction As shown in [3], [8], forest reflectivity profiles p can typically be approximated as a linear combination of a few basis functions: p ≈ Ψα (3) where Ψ ∈ RNz×Nα is the dictionary whose columns are the basis functions and α ∈ RNα is the vector of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' For an adequately chosen basis, only a few functions at a time are necessary to approximate a given profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The vector α is then sparse: most coefficients in α are zero and the number of non-zeros ∥α∥0 is small, as is the L1 norm ∥α∥1 which is often used as a proxy to measure sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The reflectivity profile �p(CS) = Ψ�α can be estimated using the following minimization problem [8]: �p(CS) = arg min p≥0 ∥Adiag(p)AH − �Σ∥2 F + λ∥Ψ†p∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (4) where λ is a hyper-parameter responsible for balancing the weight of the sparsity constraint, with respect to the data- fidelity term defined by the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The pseudo inverse matrix, Ψ†, in the L1 norm term verifies ΨΨ†p = α, and may be replaced with ΨH in the case of a unitary orthog- onal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Solving this problem typically requires several tens or hundreds of iterations of a minimization algorithm and the proper tuning of the hyper-parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Rather than using a linear model for the profiles p, non- linear models such as the generative model discussed in paragraph II-D can be used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=', Gaussian mixture models, or exponential profiles [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' However, inverting such models can be difficult due to identifiability issues and require additional constraints such as order constraints as recently proposed in a different modality of remote sensing for the estimation of phenological parameters from NDVI time series [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Our early experiments have shown that it was preferable to jointly learn a non-linear model for p = m(α) and an estimator �α by training a deep neural network with an encoder-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The code α in the latent space learnt by the en- coder is then turned into a profile by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The use of a simple feed-forward pass of a light-weight neural architecture leads to an extremely efficient tomographic inversion method, as described in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Proposed method The different steps of our method are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Starting from a ground truth profile created with a defined generative model, SAR measurements y can be simulated using the steering matrix A of a specific geometric confi- guration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' To ease the network training, we propose to provide as input to the network the beamforming profile computed from a multi-look correlation matrix �R obtained from L measurements y1 to yL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' In this way, the network input and output are in the same space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' both positive-valued vectors, easier to manipulate than complex-valued vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Besides, the beamforming profile depends less on the interferometric baselines than the correlation matrix �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It includes the physics of acquisition through the steering matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The correlation matrix has been used rather than the co- variance matrix �Σ to make the method invariant to differing intensity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The final reconstruction will therefore be �p = Tr(�Σ/N) ˜p, with ˜p the output of the network, to recover the actual intensity value of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' We have chosen to use a simple network architecture with an encoder-decoder going through a latent space of reduced dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The choice of this dimension is made to represent the number of parameters describing the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The loss used is a quadratic loss between the ground-truth simulated profile and the output profile, a standard choice for regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Generative model The distribution of reflectivity in a forest area cell is mainly composed of two peaks, one for the ground and one for the canopy, as validated by the latest tomographic reconstruction methods [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' These peaks can be represented by many SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Pipeline of the data simulation, training and testing process of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' basis functions, of which the simplest case is that of two Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The proposed approach therefore aims at learning a latent representation of a forest reflectivity profile from simulated profiles composed of two Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The parameters of these Gaussians are adapted to the type of forests observed in our SAR datasets for each training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' boreal forests in L band and tropical forests in P band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The ranges of these parameters are chosen to be large, to cover all possible profiles met in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The influence of the choice of these parameters is discussed in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Simulated data 10000 simulated profiles sampled on Nz uniformly spaced heights were generated with a mixture of two Gaussians with various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The corresponding covariance ma- trices Σ = Adiag(p)AH were built with a steering ma- trix A randomly selected among steering matrices of an actual tomographic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Circular complex Gaussian sam- ples {yl}l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='.L were then drawn according to these covari- ance matrices (by multiplying a white speckle noise wl by Adiag(√p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Sample correlation matrices �R were computed from each set of L samples yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The beamforming profiles were then reconstructed: ˜p(B) i = aH(zi) �Ra(zi) N 2 (5) where a is a steering column vector of the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' These beamforming profiles were then used as input by the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The trained encoder consists of 4 linear layers with a decreasing number of neurons (starting with Nz = 512 and encoding the input data in a latent space of size 5) and a symmetric decoder of 4 linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The depth of the network is voluntarily kept small and has been empirically validated, as discussed in section III-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Each layer is unbiased and has a leaky ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Training was performed on the dataset consisting of the simulated beamforming profiles, divided into training (75%) and validation (25%) sets, in mini- batches of size 32 with an Adam optimizer and a learning rate of 10−3 for 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' In the following subsections, the results presented were computed using the steering matrices of the BioSAR-2 cam- paign, more thoroughly presented in section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 1) Analysis of the reconstructed profiles: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 2 shows, for two different simulated profiles, the reconstructions obtained with the beamforming algorithm, Capon filter, wavelet-based CS and our method for 100 speckle realizations in the measure- ment simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' As we are interested in comparing the shapes of the profiles reconstructed by each algorithm and since they do not compute the same physical quantity, the methods corresponding to the output power of a filter have been plotted above and the reconstructed and reference reflectivity profiles below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The left profile composed of two large Gaussians favors the spectral estimation methods and shows that the wavelet- based CS fails to reconstruct large lobes, while the profile on the right simulates very narrow responses, more suitable for the latter but only roughly reconstructed by Beamforming for the level of noise considered (L = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' As for the Capon filter, it performs well with low thermal noise, but is very affected when the noise increases, even with suitable diagonal loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The neural network produces profiles with a width that follows more closely the ground-truth profiles in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (a) Beamforming (blue) Capon (green) (b) Beamforming (blue) Capon (green) (c) Ground-truth (black) Proposed method (red) Wavelet-based CS (magenta) (d) Ground-truth (black) Proposed method (red) Wavelet-based CS (magenta) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Reconstruction of a profile consisting of wide (left) and narrow (right) Gaussians with Beamforming, Capon, Wavelet-based CS and our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' For each approach, the average profile over 100 measurements with various speckle realizations and its interquartile range have been plotted, with the reference profile shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 2) Architecture choice: Different latent space sizes were tested to support the intuition of keeping a size corresponding le-1 4 0 10 0 10 20 301 0 10 0 10 20 301e-2 2 0 10 0 10 20 30le-1 1 0 10 0 10 20 30SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' JANUARY 2023 4 TABLE I MEAN AND STANDARD DEVIATION OF THE MEAN SQUARED ERROR (MSE) VALUES AFTER 20 TRAININGS DURING 200 EPOCHS FOR EACH DIFFERENT LATENT SPACE SIZES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' COMPARED TO A NORMALIZED ERROR BETWEEN THE INPUT BEAMFORMING AND REFERENCE PROFILE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Latent space size 3 4 5 6 8 10 15 20 MSE (×10−1) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='13 to the number of parameters needed to simulate a mixture of two Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The number of looks L was set to 100 for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The reconstruction errors, compared to a normalized error between the input beamforming �p(B) and reference profile p, ∥�p(B) ⟨�p(B) , p⟩ ⟨�p(B) , �p(B)⟩ −p∥2 2, where ⟨· , ·⟩ is the L2 scalar product, are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' They indicate that a latent space of size smaller than 5 significantly increases the reconstruction error, while a larger latent space brings only marginal improve- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Boreal forest at L band Testing was first performed on a tomographic stack of 6 airborne L-band SAR images of a boreal forest in northern Sweden, acquired during the BioSAR-2 campaign led by the DLR in 2008 [15], with a vertical resolution varying from 6 m in near range to 25 m in far range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' A local window of around 60 looks is used to compute the covariance matrix used by the beamforming algorithm to compute the tomograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 3 shows an example of a tomogram in HH polarization reconstructed with the proposed method, after compensation of the topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It is compared to the results obtained with beamforming, Capon and the wavelet-based CS method developed in [8], showing that the trained network does indeed improve the resolution of the tomogram compared to spectral estimation methods, while maintaining a representative volume of the tree crown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This is not the case with the wavelet-based CS, which reconstructs narrow peaks and can therefore predict several peaks when this volume is large, depending on the choice of the regularization parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The orders of magnitude of the computational time required to compute these tomograms are given in Table II (all compu- tations are done on one CPU, even the network training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The wavelet-based CS approach was computed using the optimi- sation library CVX and the default solver SDPT3 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' These statistics highlight one of the major advantages of using deep learning to reconstruct forest reflectivity profiles, which is the gain in time of this method compared to classical optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This feature will be even more important when scaling up and applying tomographic reconstruction on an entire SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Tropical forest at P band Other tests were performed on airborne tomographic SAR data acquired at P band by the ONERA over the test site of Paracou in French Guiana, during the TropiSAR campaign in 2009 [17], also comprising 6 tracks, with a vertical resolution of around 15 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The correlation matrix used to compare the different methods is calculated with 56 looks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' A sample of the results and comparisons are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The proposed (a) Beamforming (b) Capon (c) Wavelet-based CS (d) Proposed method (deep learning) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Tomogram for a specific azimuth value in a boreal forest at L band derived with the method: (a) Beamforming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (b) Capon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (c) Wavelet-based CS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (d) Predicted by the proposed neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' TABLE II ORDER OF MAGNITUDE OF THE COMPUTATION TIME FOR THE RECONSTRUCTION OF A TOMOGRAM OF SIZE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='4 KM × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content='6 M ON A CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Method Computation time (s) Beamforming 2 Capon 3 Wavelet-based CS (CVX) 1500 Proposed method � Training 200 Inference 3 approach here seems even more adapted to tropical than boreal forests, providing high resolution reconstructed profiles with fewer information loss than with the wavelet-based CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' DISCUSSION During training, the network is fed with beamforming pro- files computed from samples generated according to a model of covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This model, defined in equation (2), neglects several phenomena such as the integration of back- scattered complex amplitudes produced by scatterers located in the neighborhood according to the SAR impulse response, or the temporal decorrelation between acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It could be refined, but our results on real data already show a good 30 height z [m] 20 10 10 20 0 30 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 40 200 400 600 800 range [m]30 w] 20 10 N height 2 10 20 0 30 10 40 200 400 600 800 range [bin]30 0 [w] 20 10 height z [ 10 20 0 30 10 40 200 400 600 800 range [m]30 0 height z [m] 20 10 10 20 0 30 10 40 0 200 400 600 800 range [m]SUBMITTED TO IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, JANUARY 2023 5 (a) Beamforming (b) Capon (c) Wavelet-based CS (d) Proposed method (deep learning) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Tomogram for a specific azimuth value in a tropical forest at P band derived with the method: (a) Beamforming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (b) Capon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (c) Wavelet-based CS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' (d) Predicted by the proposed neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' generalization ability of the network trained with this simple covariance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' For the training, we used the steering matrix of a specific campaign to simulate measurements and beamforming profiles that match at best the actual SAR data considered at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The network is thus trained for a given geometric configuration and the efficiency of the network may be affected when it differs, in which case a retraining would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' However, this step is relatively inexpensive in terms of computational time (a few minutes), and tomographic data acquisition con- figurations in a given spectral band vary only slightly due to hardware constraints and the possibility of ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' In our experiments, the network was trained to reconstruct profiles composed of two Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It cannot therefore per- fectly recover a profile with a single or an extra peak, even with input profiles composed of three very distinct Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' The use of a refined simulation model could further improve the reconstruction quality, which encourages current research on a model representative of different forest types [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' CONCLUSION In this study, we explored the possibility of using deep learning approaches to improve the performance of the to- mographic inversion task on forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Tests on real data show promising results, both in terms of quality and in terms of the computation time needed to reconstruct a large image, with an overhead for the feedforward pass negligible com- pared to the computation of the beamforming profile and an acceleration by several orders of magnitude with respect to iterative regularized inversion algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' This will be essential for the systematic processing of future data from the ESA BIOMASS [18] mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It confirms the strong potential of the application of deep learning in this field to super-resolve existing reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' It could also be interesting to consider directly using the measured intensities and acquisition geometry as input to the neural network, to avoid the need to train a different network for each tomographic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' REFERENCES [1] L.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Davidson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Petit, “The TropiSAR air- borne campaign in French Guiana: objectives, description, and observed temporal behavior of the backscatter signal,” IEEE Transactions on Geoscience and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 50, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 3228–3241, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Fletcher and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' Rider, “Report for mission selection: an earth explorer to observe atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' BIOMASS.” ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' ESA SP-1324/1 (3 volume series), European Space Agency, Noordwijk, The Netherlands, 5 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} +page_content=' 60 0 height z [m] 40 10 20 20 0 30 20 40 0 250 500 750 1000 range [m]60 0 height z [m] 40 10 20 20 0 30 20 40 0 250 500 750 1000 range [bin]60 0 height z [m] 40 10 20 20 0 30 20 40 0 250 500 750 1000 range [m]60 0 height z [m] 40 10 20 30 20 40 0 250 500 750 1000 range [m]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FAT4oBgHgl3EQfjh0a/content/2301.08605v1.pdf'} diff --git a/wtFIT4oBgHgl3EQfzSt7/content/tmp_files/2301.11364v1.pdf.txt b/wtFIT4oBgHgl3EQfzSt7/content/tmp_files/2301.11364v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..810fa0058a5cc84a3852f796b21946bf8845091c --- /dev/null +++ b/wtFIT4oBgHgl3EQfzSt7/content/tmp_files/2301.11364v1.pdf.txt @@ -0,0 +1,2986 @@ +Prepared for submission to JHEP +Describing metric-affine theories anew: alternative +frameworks, examples and solutions +Damianos Iosifidisa,b Konstantinos Pallikaris,a,1 +aLaboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 +Tartu, Estonia +bInstitute of Theoretical Physics, Department of Physics Aristotle University of Thessaloniki, 54124 +Thessaloniki, Greece +E-mail: damianos.iosifidis@ut.ee, konstantinos.pallikaris@ut.ee +Abstract: In this work we describe metric-affine theories anew by making a change of field +variables. A series of equivalent frameworks is presented and identifications are worked out +in detail. The advantage of applying the new frameworks is that any MAG theory can be +handled as a Riemannian theory with additional fields. We study the Hilbert-Palatini action +using the new field variables and disclose interesting symmetries under SO transformations +in field space. Then, we use solvable and suitable Riemannian theories as seed models for +solvable MAG theories, restricting ourselves to three examples. We present a black hole +solution with torsion and non-metricity which under a certain tuning acquires a regular +core. A de Sitter universe with the expansion powered by 3-form torsion, is also reported. +1Corresponding author. +arXiv:2301.11364v1 [gr-qc] 26 Jan 2023 + +Contents +1 +Introduction +1 +2 +Preliminaries +3 +3 +The alternative framework +5 +4 +Revisiting the Hilbert-Palatini action +9 +4.1 +FF vs. AF +9 +4.2 +Projective symmetry and the AF◦ +12 +4.3 +SO(1, 2) symmetry, torsion/non-metricity rotations, and the d-AF◦ +14 +5 +Exciting the connection: a series of examples +16 +5.1 +The MAGswell theory +17 +5.2 +Non-linear interacting MAG theory, black holes and solitons +22 +5.3 +Cosmological constant powered by torsion +27 +6 +Summary and future prospects +28 +A Irreducible decomposition of a rank-3 tensor +30 +B Glossary +31 +1 +Introduction +The general theory of relativity is perhaps one of the most elegantly simple theories of +physics with such a strong impact. The ingenious statement that matter tells space-time +how to curve, and curved space-time tells matter how to move, crystallized into Einstein’s +equations, +Gµν = 8πG +c4 Tµν, +along with the many experimental tests the theory has successfully passed since its birth, +have truly established it as the most widely accepted theory of gravitation. +Despite its successes, Einstein’s theory has its shortcomings. To mention but a few of +them, first, general relativity is not a perturbatively renormalizable quantum field theory +meaning that it gets striped of its predictive power at high energies with the Planck mass +constituting the cut-off scale. Second, the small measured value of the cosmological constant +leads to a perplexing discrepancy between theory and experiment, a naturalness problem +known as the cosmological constant problem (see [1] for an extensive review). Prominent +shortcomings are also the flatness and horizon problems [2–5] plaguing the standard model +of hot big bang cosmology, which are properly addressed in cosmic inflation [6]. +– 1 – + +In view of the above, looking for alternative gravity theories is a justified course of +action.1 The search for these so-called modified theories of gravity is in essence a search +for healthy field equations that differ from those of Einstein. Owing to Lovelock and his +undisputed theorem [10, 11], there is a list of assumptions that we need to break (in one +or more ways) in order to find such a set of equations. In particular, one of the assump- +tions is that space-time is a smooth Lorentzian manifold equipped with a time orientation. +Therefore, we can dodge the stringent consequences of the theorem by permitting the affine +connection to be an independent field variable beyond the metric. +A general connection has both torsion and non-metricity, and a gravitational theory for +the metric and the affine connection is known as a Metric-Affine Gravity (MAG) theory [12]. +MAG theories exhibit many attractive features. First, the presence of a new gravitational +potential, the independent affine connection, brings gravity conceptually closer to the other +interactions whose mediators are gauge connections.2 +Second, an intriguing feature of +metric-affine theories of gravity is the emergence of a hypermomentum current [13–16] in +the presence of matter couplings to the gauge connection. +This differential form, obtained by varying the matter action with respect to the gauge +connection, can be decomposed into the irreducible spin, dilation and shear parts which +ought to excite the post-Riemannian structure. In the above sense, MAG theories bring +forth an astonishing interplay between matter with non-trivial microstructure and non- +Riemannian effects. Finally, note that an interesting discussion has been revived about the +status of MAG as a quantum theory though definitive conclusions are yet far from being +drawn (see [17–19] and references therein).3 +As with Riemannian theories, one is particularly interested in MAG theories which are +solvable, ideally exactly solvable (at least for some symmetry ansatz). If the task of finding +exact solutions in Riemannian theories is in most cases a difficult one, then the trouble +gets double in MAG because we also have to determine the connection. In fact, perhaps +the most persistent obstruction to obtaining exact solutions with non-vanishing torsion and +non-metricity in metric-affine theories,4 is the computational complexity one is bound to +face when attempting to solve the field equations for the affine connection. +In the dominant part of the MAG literature, the strategy to make the connection +dynamical is, roughly put, to consider (at least) quadratic curvature invariants like R2, +RµνRµν, RλρµνRλρµν, et cetera. This strategy is indeed well-motivated and fairly general, +but it can quickly turn any attempt at finding a solution into a nearly impossible task, even +for relatively simple (in form) Lagrangians of this sort. The reason behind this is that the +affine connection is a very compact package of a large number of degrees of freedom, the +dynamics of which are encoded in the components of a single tensor equation and presented +in an awfully coupled manner. In fact, among other techniques, one almost always tries to +split this master tensor equation into simpler, hopefully decoupled equations by acting on +1For a review of the zoo of modified-gravity theories see [7–9] and references therein. +2See the notion of affine gauge theory in [12]. +3See also [20–34] for some recent advances in the field. +4See [35–39] and references therein for some examples of black hole solutions with torsion and/or non- +metricity. +– 2 – + +it with some symmetry projector, or taking traces. Therefore, it may not always be the +case that the affine connection is the optimal field variable, beyond the metric, to describe +a MAG theory, at least not for all intents and purposes. +In this work, we embrace this point of view and use it as a motivation for our proposal. +Our goal is to make a change of field variables that will allow us to trade the connection +field equations for an equivalent “decongested” system of simpler field equations obtained +by letting an action vary with respect to tensor and vector fields. These tensor and vector +fields, used to describe MAG theories anew, will be the irreducible pieces of torsion and +non-metricity under the Lorentz group. Ipso facto, they are identified with the fundamental +fields, the metric and the affine connection. We then work out a complete mapping between +the two frameworks which can be later used as a dictionary. The advantage of the new +framework is that we can now handle MAG theories as Riemannian theories with additional +fields, at least within the context of the variational problem. These additional fields are +part of the space-time geometry it self, and not some external entities. +With the new framework established, we proceed with giving examples of how to con- +struct MAG theories in vacuum which result in a selective and tractable self-excitation of +the connection. Although there is no universal prescription, a basic idea underlies all our +examples. We take Riemannian theories with additional fields (vectors and tensors), which +we exactly know how to solve, and we cast them, after some minor necessary modifications, +into MAG theories which effectively yield the same field equations. The role of the addi- +tional fields is now performed by the new field variables. Their propagation is tantamount +to the excitation of (part of) the post-Riemannian structure. Even though the form of the +metric solution in such MAG theories will, more or less, be already known in the gravity +literature, the full solution, including the connection, will be novel, for it will in general +feature non-zero torsion/non-metricity backgrounds. +Plan of this work. +In section 2 we convey the bare minimum in metric-affine theories. +Then, in section 3 we present the alternative framework and a detailed mapping between +the latter and the ordinary Palatini approach. Using the new framework, we revisit the +Hilbert-Palatini action in section 4 hoping for fresh insight, and we introduce a useful +variant of the new framework when projective symmetry is at play. Finally, in section 5 we +showcase a series of examples where we apply the previously developed frameworks, and +we also report solutions therein, concluding in section 6. +2 +Preliminaries +This section is devoted to a brief communication of the MAG preliminaries. In metric-affine +theories the affine connection is an independent field variable beyond the metric. We use +it to define a covariant derivative whose action on vector and co-vector fields is given by +∇µV ν = ∂µV ν + Γν +λµV λ, +(2.1a) +∇µVν = ∂µVν − Γλ +νµVλ, +(2.1b) +– 3 – + +where Γλ +µν are the connection symbols. A general affine connection features both torsion +and non-metricity given by +T λµν = 2Γλ +[νµ], +(2.2a) +Qλµν = −∇λgµν, +(2.2b) +respectively. The former introduces twisting; parallel transport along a closed path results +in a translation. The latter measures the failure of the metric to be covariantly constant; +parallel transport brings about a change in vector norms. +Out of torsion and non-metricity we can construct three vectors and one axial tensor. +Regarding torsion, we have the vector Tµ = T λµλ and the axial tensor +Sα1...αn−3 = − +1 +6(n − 3)!˜ϵα1...αn−3λµνTλµν. +(2.3) +Here, ˜ϵα1...αn = √−gϵα1...αn with ϵα1...αn being the Levi-Civita symbol in n space-time di- +mensions. Our convention for the symbol is ϵ01...n−1 = 1 = −ϵ01...n−1. In n = 4 dimensions, +the above axial tensor is known as the torsion pseudo-vector, +Sα = −1 +6˜ϵαλµνTλµν. +(2.4) +Regarding non-metricity, we have the vector Qµ = Qµαβgαβ, which is proportional to what +is often called the Weyl vector in MAG lore, and ˇQµ = Qαβµgαβ. +Continuing, we define the curvature tensor of the general affine connection as +Rµναβ = ∂αΓµ +νβ + Γµ +ραΓρ +νβ − α ↔ β. +(2.5) +From the above we can form three independent contractions, +Rνβ = Rµνµβ, +(2.6a) +ˆRαβ = Rµµαβ = ∂[αQβ], +(2.6b) +ˇRλα = Rλµανgµν, +(2.6c) +which go by the name Ricci tensor, homothetic-curvature tensor, and co-Ricci tensor, re- +spectively. Notice that only the last contraction requires a metric. Finally, contracting +indices once more with the metric, we form the Ricci scalar R = Rµνgµν = ˇRµµ. As per +tradition, we will refer to the curvature tensor associated with the Levi-Civita connection as +the Riemann tensor. Its single (double) trace will bear the name Riemannian Ricci tensor +(scalar). +Furthermore, it is a well-established fact that every affine connection differs from an- +other affine connection by a tensor. Therefore, we can always write a general affine connec- +tion as +Γλ +µν = ˜Γλ +µν + Nλµν, +(2.7) +where +˜Γλ +µν = 1 +2gρλ (∂µgνρ + ∂νgµρ − ∂ρgµν) +(2.8) +– 4 – + +are the Christoffel symbols, and +Nλµν = 1 +2gρλ (Qµνρ + Qνρµ − Qρµν − Tρµν − Tνµρ − Tµνρ) +(2.9) +is the so-called distortion tensor encompassing the non-Riemannian DoF. Torsion and non- +metricity can always be traded for the distortion tensor via the relations T λµν = −2Nλ[µν] +and Qλµν = 2N(µν)λ. +Note that eq. (2.7) suggests that we can split off any quantity into a Riemannian part +and non-Riemannian contributions; this is the reputed post-Riemannian expansion of a +quantity. For instance, the post-Riemannian expansion of the curvature tensor reads +Rµναβ = ˜Rµναβ + 2 ˜∇[αNµ +|ν|β] + 2Nµ +λ[αNλ +|ν|β], +(2.10) +where ˜∇α is the Levi-Civita covariant derivative and ˜Rµναβ the Riemann tensor. Unless +otherwise stated, quantities with a tilde accent will always stand for objects associated with +the Levi-Civita connection. +3 +The alternative framework +Observe that the presence of an affine connection as an independent field variable introduces +n3-many additional a priori DoF. Undeniably, the affine connection, being an essential +constituent of the metric-affine geometry, is a meaningful variable to work with; torsion +and non-metricity are after all properties of a connection. However, squashing that many +degrees into a single field is not always the most convenient option. In this section, we +instead distribute them among seven fields which correspond to the irreducible pieces of +torsion and non-metricity. This strange way of re-organizing the connection DoF will be +suitable for purposes presented during a later stage. +The new fields will of course be identified with the metric and the affine connection, +the fundamental field variables in metric-affine theories, thus allowing us — via this change +of field variables — to describe any MAG theory anew. We will show in full generality that +the field equations derived within this new framework imply and are implied by the field +equations obtained in the familiar context of the Fundamental (or Palatini) Framework (FF) +where the metric and the affine connection are the independent variables. The freedom to +switch between different formulations of the same theory will prove to be a great asset in +the next sections. +In what follows, ˚aλµ... denotes the completely traceless part of a tensor aλµ..., whereas +¯aλµ... denotes the complement of ˚aλµ... in aλµ..., viz., ¯aλµ... = aλµ... −˚aλµ.... The irreducible +decomposition of the torsion tensor under the Lorentz group yields +Tλµν = Hλµν +˚tλµν + ¯tλµν, +(3.1) +where +Hλµν = T[λµν], +(3.2a) +˚tλµν = Tλµν − Hλµν − ¯tλµν, +¯tλµν = +2 +n − 1gλ[νTµ]. +(3.2b) +– 5 – + +Note that instead of the 3-form field Hλµν one may alternatively use the dual tensor +Sα1...αn−3 defined in (2.3). +Similarly, for non-metricity we have +Qλµν = ˚πλµν + ¯πλµν + ˚qλµν + ¯qλµν, +(3.3) +where +˚πλµν = Q(λµν) − ¯πλµν, +¯πλµν = +1 +n + 2g(λµρν), +(3.4a) +˚qλµν = Qλµν − πλµν − ¯qλµν, +¯qλµν = +2 +3(n − 1) +� +gλ(µuν) − gµνuλ +� +, +(3.4b) +uµ = ˇQµ − Qµ, +ρµ = 2 ˇQµ + Qµ. +(3.4c) +Using the defining eqs. (2.2), equations (3.2) and (3.4) tell us how to express the irre- +ducible pieces in terms of the metric and the affine connection. The other way around, +eqs. (3.1) and (3.3) tell us how to express the affine connection in terms of the metric and +the irreducible pieces using eqs. (2.7), (2.8), and (2.9). +Since we will work with many fields, we find it befitting to use multi-field notation. Let +us introduce two objects, O and A, with components ON +λµν and AI +µ, respectively. They are +given by +Oλµν = {Hλµν, tλµν, πλµν, qλµν} , +(3.5a) +Aµ = {Tµ, ρµ, uµ} . +(3.5b) +Einstein’s summation convention will also be adopted for indices M, N, ..., which take values +in {1, 2, 3, 4}, and for indices I, J, ..., which take values in the subset {2, 3, 4}.5 We can +lower/raise these indices with the reference metrics δMN and δIJ, respectively. As above, +whenever the capital indices are omitted, the objects should be understood as column +vectors in Euclidean space. Finally, the term Alternative Framework (AF) will be coined +for the formulation of a MAG theory in terms of the set {g, ˚ +ON, AI} of field variables. +With all the necessary ingredients at our disposal, let us consider a general n-dimensional +MAG action in the FF, say +I[g, Γ] = +� √−gdnxL, +(3.6) +where g ≡ det g. We let it vary in order to get +δI = +� √−gdnx +� +Eµνδgµν + ∆λµνδΓλ +µν +� ++ s.t., +(3.7) +where s.t. +denotes the surface terms arising from integrating by parts. +We have also +abbreviated the functional derivatives as +Eµν = +1 +√−g +δI +δgµν , +∆λµν = +1 +√−g +δI +δΓλµν +(3.8) +5Note the use of slanted numerals for the value of an internal index as opposed to µ, ν, ... = 0, ..n − 1. +– 6 – + +The field equations read +Eµν = 0, +∆λµν = 0, +(3.9) +with Eµν being a symmetric tensor.6 +On the other hand, considering eqs. (2.7), (2.8), (3.1), and (3.3), we can write the +previous action in the AF, namely +I +� +g, ˚ +ON, AI� += +� √−gdnxL. +(3.10) +Letting I vary we get +δI = +� √−gdnx +� +ˆEµνδgµν + ˚ +Oλµν +N δ˚ +ON +λµν + Aµ +I δAI +µ +� +(3.11) +plus surface terms where ˚ +ON and ˚ +ON belong to the same irreducible tensor subspace as +Lorentz tensors. The field equations read +ˆEµν = 0, +˚ +Oλµν +N += 0, +Aµ +I = 0, +(3.12) +where ˆEµν is a symmetric tensor. Observe that the traceless property of ˚ +ON +λµν must be +preserved when the action is varied. +This condition can be enforced with a Lagrange +multiplier. +The result is equivalent to simply demanding that the functional derivative +with respect to ˚ +ON +λµν, ˚ +Oλµν +N +, must be traceless. +With the above in hand, we turn our attention to finding the identities relating the +functional derivatives { ˆE, ˚ +ON, AI} to {E, ∆}. These identities will arise via identifications. +Expressing eq. (3.7) in terms of the AF variables, one finds that +˚ +Oλµν +1 += −1 +2∆[λµν], +˚ +Oλµν +2 += ˚ +D[µν]λ, +(3.13a) +˚ +Oλµν +3 += 1 +2 +˚∆(λµν), +˚ +Oλµν +4 += −˚ +Dλ(µν), +(3.13b) +Aµ +2 = +1 +n − 1 +� +∆µλλ − ∆λµλ� +, +(3.13c) +Aµ +3 = +1 +6(n + 2) +� +∆λλµ + ∆λµλ + ∆µλλ +� +, +(3.13d) +Aµ +4 = +1 +3(n − 1) +� +2∆µλλ − ∆λλµ − ∆λµλ� +, +(3.13e) +where ˚ +D and ˚∆ are given in eqs. (A.2) of the appendix. Finally, we also have +ˆEµν = Eµν − +∆α(µν) − δα +(µ∆ν)ββ +n − 1 +� n − 1 +6(n + 2)ρα + Tα + 2 +3uα +� ++ 2∆[αβ]β +n − 1 +˚t(µν)α − +−2∆(αβ)β + ∆ββα +2(n + 2) +˚πµνα + ∆αββ − ∆β(αβ) +n − 1 +˚qαµν − +−1 +2∆(µ +αβ � +Hν)αβ − ˚πν)αβ + 2˚qν)αβ + 2˚t|βα|ν) +� ++ ++1 +2 +� +˜∇α∆(µν)α + ˜∇α∆(µ|α|ν) − ˜∇α∆α(µν) +� +, +(3.14) +6The delicate issue of surface-term handling is out of the scope of this paper. We rather assume that +one has by all means ensured that the variational problem is well-posed. +– 7 – + +where we used the identities +˚t[µν]λ = −1 +2 +˚tλµν, +˚q(µν)λ = −1 +2˚qλµν. +(3.15) +If we let the fields ˚ +ON and AI on the right hand side of eq. (3.14) denote expressions +involving the metric and the connection symbols (see eqs. (3.2) and (3.4)), the above simply +gives us ˆE in terms of the FF quantities. +At this stage, we find it useful to display the “inverted form” of eqs. (3.13) by expressing +∆ in terms of ˚ +ON and AI. Using the identity +˚ +D[µν]λ = ˚ +D[µ|λ|ν] − ˚ +Dλ[µν], +(3.16) +we directly obtain +˚∆[λµν] = −2˚ +Oλµν +1 +, +˚∆(λµν) = 2˚ +Oλµν +3 +, +(3.17a) +˚ +Dλ[µν] = −2 +� +˚ +O[µν]λ +4 ++ ˚ +Oλµν +2 +� +, +˚ +Dλ(µν) = −˚ +Oλµν +4 +. +(3.17b) +The last three equations in (3.13) form a separate matrix subsystem, invertible for n > 1, +whose inversion yields +∆λλµ = (n − 1) +� +Aµ +2 − 2Aµ +4 +� ++ 2(n + 2)Aµ +3, +(3.18a) +∆λµλ = (n − 1) +� +Aµ +4 − Aµ +2 +� ++ 2(n + 2)Aµ +3, +(3.18b) +∆µλλ = (n − 1)Aµ +4 + 2(n + 2)Aµ +3. +(3.18c) +Recalling that the irreducible decomposition of a general rank-3 tensor has the form (A.1), +it all boils down to the equation +∆λµν/2 = ˚ +Oλµν +3 +− ˚ +Oλµν +1 ++ ˚ +Oνµλ +4 +− ˚ +Oλµν +2 ++ ++3A(λ +3 gµν) + gν(λAµ) +4 − gµλAν +4 + gλ[µAν] +2 . +(3.19) +Finally, one may take the above result, plug it into eq. (3.14), and write the latter as +Eµν = ˆEµν + ..., +(3.20) +which provides us with E in terms of AF quantities. Finally note that the vanishing of a +tensor implies that all its irreducible pieces vanish separately, and vice versa. +Let us now mold all these technical details into the main result we wish to convey. +When the field eqs. (3.12) hold true, we have that ∆λµν = 0 via eq. (3.19) and Eµν = 0 +via eq. (3.20), ergo, the field eqs. (3.9) are implied. The other way around, when the field +eqs. (3.9) hold true, we have ˚ +Oλµν +N += 0 and Aµ +I = 0 via eqs. (3.13). From eq. (3.14), it +further follows that ˆEµν = 0, ergo, the field eqs. (3.12) are implied. +Consequently, we +have shown an equivalence relation in detail, in particular, that the field equations in the +two formulations imply and are implied by each other. We also remark that, having the +field equations in one of the two frameworks, it is always possible to reconstruct the field +equations in the other. +– 8 – + +Lastly, having set up the new framework, we find it useful to report an interesting +correspondence. +There exist certain linear connection transformations in the FF which +amount to translations of only one irreducible piece at a time (preserving the rest) in the +AF. Before disclosing them, let us bring yet another pair of multi-fields to our aid, o and +a, with components oN +λµν and aI +µ, respectively. Note that ˚oN and ˚ +ON belong to the same +irreducible tensor subspace as Lorentz tensors. After some straightforward algebra we arrive +at a 1:1 correspondence between the translations +˚ +O′N = ˚ +ON +˚oN, +A′I = AI + aI, +(3.21) +in the AF (space-time indices understood, thus omitted) and the linear connection trans- +formations Γ′λ +µν = Γλ +µν + (δΓ)λ +µν with +(δΓ)λµν = −1 +2˚o1 +λµν, +(δΓ)λµν = −˚o2 +νµλ, +(3.22a) +(δΓ)λµν = 1 +2˚o3 +λµν, +(δΓ)λµν = −˚o4 +λµν, +(3.22b) +(δΓ)λµν = +2 +n − 1gν[µa2 +λ], +(δΓ)λµν = +1 +2(n + 2)a3 +(λgµν), +(3.22c) +(δΓ)λµν = +2 +3(n − 1) +� +gµνa4 +λ − gλ(µa4 +ν) +� +, +(3.22d) +in the FF. +We also report that under a local Weyl re-scaling of the metric, g′ +µν = e−2φ(x)gµν, +the tensor fields ˚ +ON +λµν must have conformal weight −2, and thus, transform as the metric, +whereas +T ′ +µ = Tµ, +ρ′ +µ = ρµ + 2(n + 2)∂µφ, +u′ +µ = uµ − 2(n − 1)∂µφ. +(3.23) +Clearly, the combination ρµ +(n+2)uµ/(n−1) is itself a Weyl invariant. It corresponds to +3(n ˇQµ − Qµ)/(n − 1) in the FF. We shall now proceed with a highly pedagogical example. +4 +Revisiting the Hilbert-Palatini action +4.1 +FF vs. AF +The n-dimensional Hilbert-Palatini (HP) action reads +IHP = 1 +2 +� √−gdnxR, +(4.1) +in units ℏ = c = MPl = 1 where MPl is the reduced Planck mass. This is the standard FF +action which is invariant under the so-called projective transformation +Γ′λ +µν = Γλ +µν + +1 +n − 1δλ +µξν, +(4.2) +with ξµ being an arbitrary vector field. +– 9 – + +The field equations in the FF read +2Eµν ≡ R(µν) − 1 +2Rgµν = 0, +(4.3a) +2∆λµν ≡ δν +λNµαα − Nµνλ − Nνλµ + Nαλαgµν = 0. +(4.3b) +We chose to express the connection field equations in terms of the distortion tensor in order +to achieve a more compact output. The invariance of eqs. (4.3) under (4.2) can be easily +seen from the fact that +R′ +µν = Rµν + +2 +n − 1∂[µξν], +∆λλµ ≡ 0, +(4.4) +the right one holding true identically (off-shell). +It is a well-known fact that the solution to ∆λµν = 0 is the affine connection +Γλ +µν = ˜Γλ +µν + δλ +µVν, +(4.5) +where Vµ is some undetermined vector field. Since +Γλ +µν = ˜Γλ +µν + δλ +µ +� +Vν + +1 +n − 1ξν +� +(4.6) +is also a solution, we conclude that the affine connection solving the connection field equa- +tions is just the Levi-Civita connection up to the choice of gauge. The effective form of the +metric field equations becomes +˜Rµν = 1 +2 +˜Rgµν, +(4.7) +i.e., the HP action is effectively Einstein gravity. +On the other hand, in the AF, whenever we write R we just mean the expression +˜R + RT + RV + ˜∇µ (2T µ + uµ) , +(4.8) +where ˜R is the Riemannian Ricci scalar and +RT = −1 +4H2 − 1 +4˚π2 + 1 +2˚qλµν˚qλµν + 1 +2 +˚tλµν˚tλµν + ˚qλνµ˚tνµλ, +(4.9a) +RV = +n − 1 +36(n + 2)ρ2 − n − 2 +n − 1T 2 + 5 − 2n +9(n − 1)u2 + 1 +18ρµuµ − n − 2 +n − 1Tµuµ. +(4.9b) +Therefore, up to surface terms, our AF action reads +IHP = 1 +2 +� √−gdnx +� +˜R + RT + RV +� +. +(4.10) +The analogue of a projective transformation in the AF is comprised of the simultaneous +translations +A′I = AI + aI−1, +(4.11) +with +ξµ ≡ a1 +µ = +n − 1 +2(n + 2)a2 +µ = −1 +2a3 +µ. +(4.12) +– 10 – + +One can easily verify that the above transformations should only affect RV . Since it hap- +pens that RV is invariant, the transformations (4.11) constitute a symmetry of the full +action (4.10). +Now, there are two equivalent ways to proceed as we have shown in the previous section. +We can either use eqs. (4.3) to reconstruct the field equations in the AF, or we can directly +vary the integral (4.10) with respect to the AF field variables (quickest strategy). Both +methods lead to the same result, namely the field equations +˚ +Oλµν +1 +≡ −1 +4Hλµν = 0, +˚ +Oλµν +2 +≡ 1 +2 +� +˚tλµν − ˚q[µν]λ� += 0, +(4.13a) +˚ +Oλµν +3 +≡ −1 +4˚πλµν = 0, +˚ +Oλµν +4 +≡ ˚ +O(µν)λ +2 ++ 1 +8˚qλµν = 0, +(4.13b) +Aµ +2 ≡ −n − 2 +n − 1 +� +T µ + 1 +2uµ +� += 0, +(4.13c) +Aµ +3 ≡ +n − 1 +36(n + 2) +� +ρµ + n + 2 +n − 1uµ +� += 0, +(4.13d) +Aµ +4 ≡ 1 +2Aµ +2 + n + 2 +n − 1Aµ +3 = 0, +(4.13e) +and +2 ˆEµν ≡ ˜Rµν − 1 +2gµν +� +˜R + RT + RV +� +− 3 +4 +� +HµαβHναβ + ˚πµαβ˚πναβ� ++ ++˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 +2 +� +˚qµαβ˚qναβ +˚tµαβ˚tναβ� ++ ++˚tαβ(µ˚qν) +αβ − ˚qαβ +(µ˚tν)αβ −˚tβα(µ˚qαβ +ν) − +−n − 2 +n − 1TµTν + +n − 1 +36(n + 2)ρµρν + 5 − 2n +9(n − 1)uµuν − +−n − 2 +n − 1T(µuν) + 1 +18ρ(µuν) = 0. +(4.14) +It is evident that the first two lines in (4.13) suggest that ˚ +ON +λµν = 0. The remaining +two independent equations, Aµ +2 = 0 = Aµ +3, do further imply that +uµ = −2Tµ = −n − 1 +n + 2ρµ. +(4.15) +Hence, the full solution to the system (4.13) is +˚ +ON +λµν = 0, +Vµ ≡ uµ = −2Tµ = −n − 1 +n + 2ρµ, +(4.16) +where Vµ is an arbitrary vector field. Since +Tµ = −1 +2Vµ + ξµ, +ρµ = n + 2 +n − 1 (2ξµ − Vµ) , +uµ = Vµ − 2ξµ +(4.17) +is also a solution, we conclude that AI +µ = 0 up to the choice of gauge. The effective form of +the metric field equations (4.14) becomes +˜Rµν = 1 +2 +˜Rgµν, +(4.18) +i.e., the HP action in the AF is again, in effect, Einstein gravity. +– 11 – + +4.2 +Projective symmetry and the AF◦ +The careful reader would have already noticed that what is a projective symmetry in the +FF manifests itself as a true gauge symmetry in the AF. Indeed, eqs. (4.13) reveal that +there are only two independent equations for the triplet Aµ which rather signals that one +of these field variables is after all redundant. First, bear in mind that the combinations +Tµ + 1 +2uµ, +ρµ + n + 2 +n − 1uµ, +(4.19) +which are invariant under (4.11), correspond to the FF combinations +1 +2 +� ˇQµ − Qµ +� ++ Tµ, +3 +n − 1 +� +n ˇQµ − Qµ +� +, +(4.20) +respectively, which are invariant under (4.2). +Let us then discuss the idea that for n > 2, whenever the projective symmetry is at +play, one should favor a doublet +Bµ = +�2 +3 +� +Tµ + 1 +2uµ +� +, +n − 1 +9 +√ +n2 − 4 +� +ρµ + n + 2 +n − 1uµ +�� +(4.21) +over the redundant triplet Aµ. Note that the above choice of BA, where indices A, B, ... +assume values in {1, 2}, is not the most general one. Nevertheless, it is the most convenient +choice for our purposes here since it casts RV into the neat form +RV = 9(n − 2) +4(n − 1)ηABBA +µ BB +ν gµν, +(4.22) +where ηAB are the components of the two-dimensional Minkowski metric η(2) = diag(−1, 1). +The affected parts of (4.13) read +Aµ +2 ≡ −3(n − 2) +2(n − 1)Bµ +1 = 0, +Aµ +3 ≡ 1 +4 +� +n − 2 +n + 2Bµ +2 = 0, +(4.23) +whereas the metric field equations (4.14) are rendered into +2 ˆEµν ≡ ˜Rµν − 1 +2gµν +� +˜R + RT + RV +� +− 3 +4 +� +HµαβHναβ + ˚πµαβ˚πναβ� ++ ++˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 +2 +� +˚qµαβ˚qναβ +˚tµαβ˚tναβ� ++ ++˚tαβ(µ˚qν) +αβ − ˚qαβ +(µ˚tν)αβ −˚tβα(µ˚qαβ +ν) + 9(n − 2) +4(n − 1)ηABBA +µ BB +ν = 0. +(4.24) +Interestingly, when written in terms of the BA fields, the HP action and its field equations +exhibit an SO(1, 1) symmetry! Indeed, the field transformation B′ +µ = Λ(x)Bµ with +Λ = +� +cosh θ(x) sinh θ(x) +sinh θ(x) cosh θ(x) +� +, +(4.25) +preserves both of them. We remark that the manifestation of this transformation as a group +action on the field variables is exclusive to the use of the BA’s to formulate the HP action. +– 12 – + +Since the use of these field re-combinations revealed something new, we find it worth to +take a step back and generalize the whole thing to another framework which we dub AF◦ or +“diminished alternative framework”. In the AF◦, we formulate our MAG theory in terms of +the reduced set {g, ˚ +ON, BA} of field variables. The doublet Bµ should consist of two linear +combinations of AF vector fields which are invariant under (4.11). Equivalently, it should +be comprised of two linear combinations of Tµ, Qµ, ˇQµ which are invariant under (4.2), the +point being that in the AF◦ such transformations should constitute an identity operation +on our field variables. The most general combinations invariant under (4.11) are +B1 +µ = αI−1AI +µ, +B2 +µ = βI−1AI +µ, +(4.26) +with +α3 = α1 +2 + (n + 2)α2 +n − 1 +, +(4.27) +and ditto for the coefficients βA. +Let +Bµ +A = +1 +√−g +δI +δBA +µ +, +(4.28) +such that the field equations for the field BA are Bµ +A = 0. +We can directly make the +identifications +Aµ +I = αI−1Bµ +1 + βI−1Bµ +2, +(4.29) +where one has to remember that the coefficients obey the relation (4.27). Clearly, whenever +Bµ +A = 0, it follows that Aµ +I = 0. However, whenever Aµ +I = 0, it follows that Bµ +A = 0 only +when +α1β2 − α2β1 ̸= 0. +(4.30) +Therefore, the field equations in the AF◦ imply and are, under assumptions, implied by the +field equations in the AF or in the FF (if we follow the equivalence chain). +Lastly, let us see exactly how we ended up with (4.21). In terms of the fields BA, as +defined in eq. (4.26), we have that +RV = +� +f(β2 +1, β2 +2)B1 +µB1 +ν − 2f(α1β1, α2β2)B1 +µB2 +ν + f(α2 +1, α2 +2)B2 +µB2 +ν +� +gµν, +(4.31) +where +f(x, y) := +(n − 1)2x − 36(n2 − 4)y +36(n2 + n − 2)(α2β1 − α1β2)2 . +(4.32) +Moreover, the total divergence in (4.8) assumes the form +2 +α2β1 − α1β2 +˜∇µ (α2Bµ +2 − β2Bµ +1 ) . +(4.33) +To get the above, we expressed α3 in terms of α1, α2 via eq. (4.27), and ditto for the +parameters βA. Different choices for the parameters αA, βA obviously amount to different +changes of field variables. +– 13 – + +A convenient choice is one for which f(α1β1, α2β2) = 0, namely +β2 = (n − 1)2α1β1 +36(n2 − 4)α2 +, +(4.34) +provided n > 2, which yields +RV = +(n − 2)(n − 1) +(n − 1)2α2 +1 − 36(n2 − 4)α2 +2 +� +−B1 +µB1 +ν + 36α2 +2(n2 − 4) +β2 +1(n − 1)2 B2 +µB2 +ν +� +gµν. +(4.35) +Further imposing that +β1 = 6|α2| +√ +n2 − 4 +n − 1 +, +(4.36) +gives +RV = +(n − 2)(n − 1) +(n − 1)2α2 +1 − 36(n2 − 4)α2 +2 +ηABBA +µ BB +ν gµν. +(4.37) +Finally, we choose +α2 = n − 1 +18 +� +9α2 +1 − 4 +n2 − 4 , +(4.38) +for later convenience, which leads to +RV = 9(n − 2) +4(n − 1)ηABBA +µ BB +ν gµν =: RV . +(4.39) +Note that all of the above parameter choices are in agreement with (4.30) which becomes +2(n − 1) +27 +√ +n2 − 4 +̸= 0. +(4.40) +In terms of the AF fields, our new field variables, BA, read +B1 +µ = α1Tµ + n − 1 +18 +� +9α2 +1 − 4 +n2 − 4 ρµ + 1 +18 +� +�9α1 + +� +(9α2 +1 − 4)(n + 2) +n − 2 +� +� uµ, (4.41a) +B2 +µ = +� +9α2 +1 − 4 +3 +Tµ + (n − 1)α1 +6 +√ +n2 − 4 +ρµ + 1 +6 +�� +n + 2 +n − 2α1 + +� +9α2 +1 − 4 +� +uµ, +(4.41b) +where we may further fix |α1| = 2/3 so that B2 is purely a combination of traces of the +non-metricity tensor. This brings us to (4.21). Henceforth, the word AF◦ will always mean +that we use the specific doublet (4.21). +4.3 +SO(1, 2) symmetry, torsion/non-metricity rotations, and the d-AF◦ +Now, we restrict ourselves to n = 4 space-time dimensions where things get a bit more +interesting. Via the dualization (2.4), we have a pseudo-vector, and the term ∝ H2 can be +moved from RT to RV . In particular, let us introduce the objects +ˆRT = −1 +4˚π2 + 1 +2˚qλµν˚qλµν + 1 +2 +˚tλµν˚tλµν + ˚qλνµ˚tνµλ, +(4.42a) +ˆRV = 3 +2ηABBA +µ BB +ν gµν. +(4.42b) +– 14 – + +where the calligraphic indices take values in {1, 2, 3}, ηAB are the components of the +three-dimensional Minkowski metric, η(3) = diag(−1, 1, 1), and we have formed a triplet +Bµ = +� +B1 +µ, B2 +µ, Sµ +� +, +(4.43) +with BA given by (4.21). +The affected parts of (4.13) read +Aµ +2 ≡ −3(n − 2) +2(n − 1)Bµ +1 = 0, +Aµ +3 ≡ 1 +4 +� +n − 2 +n + 2Bµ +2 = 0, +(4.44) +and +˚ +Oλµν +1 +≡ 1 +4˜ϵλµναSα = 0, +(4.45) +whereas the metric field equations (4.14) are rendered into +2 ˆEµν ≡ ˜Rµν − 1 +2gµν +� +˜R + ˆRT + ˆRV +� +− 3 +4˚πµαβ˚πναβ + ++˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 +2 +� +˚qµαβ˚qναβ +˚tµαβ˚tναβ� ++ ++˚tαβ(µ˚qν) +αβ − ˚qαβ +(µ˚tν)αβ −˚tβα(µ˚qαβ +ν) + 3 +2ηABBA +µ BB +ν = 0. +(4.46) +Remarkably, when written in terms of the triplet (4.43), the four-dimensional HP action +and its field equations exhibit a larger symmetry under an SO(1, 2) group action mixing +the components BA +µ . +Of particular interest is the transformation B′ +µ = Λ(x)Bµ with +Λ = +� +� +� +1 +cos θ(x) +sin θ(x) +− sin θ(x) cos θ(x) +� +� +� , +(4.47) +which represents an SO(2) rotation in the {B2 +µ, Sµ} (field) subspace. +Specifically, the +invariance of the action under the discrete transformation with Λ(θ = π/2), constitutes an +exceptional example of a symmetry under torsion/non-metricity rotations. Indeed, omitting +the space-time indices, we have +� +� +� +B1 +B2 +S +� +� +� → +� +� +� +B1 +S +−B2 +� +� +� , +(4.48) +where Sµ is pure torsion (pseudo-vector) and B2 +µ is pure non-metricity (traces), as de- +fined in (2.4) and (4.21), respectively. Do further note that if we consider B2 +µ and Sµ as, +respectively, the real and imaginary parts of a complex vector +τµ = B2 +µ + iSµ, +(4.49) +we have that +ˆRV = −3 +2 +� +B1 +µB1 +ν − τµτ ∗ +ν +� +gµν, +(4.50) +– 15 – + +where τ ∗ is the complex conjugate of τ. The previous SO(2) symmetry now manifests itself +as a U(1) under τ ′ +µ = e−iθ(x)τµ. +As a closing remark, let us mention here that in what follows we will often use the tor- +sion pseudo-vector Sµ instead of Hλµν as a field variable in our four-dimensional projective- +invariant examples. In other words, we will be using the set {g, ˚ +OI, BA} when invoking the +AF◦, and some clarifications are in order. Letting an action +I = +� √−gd4xL[g, ˚ +OI, BA] +(4.51) +vary, we get +δI = +� √−gd4x +� +ˇEµνδgµν + ˚ +Oλµν +I +δ˚ +OI +λµν + Bµ +AδBA +µ +� ++ s.t.. +(4.52) +Therefore, the field equations read +ˇEµν = 0, +˚ +Oλµν +I += 0, +Bµ +A = 0, +(4.53) +where ˇEµν is a symmetric tensor. +Following the preceded steps, one should be able to +directly make the identifications +Bµ +1 = 3 +2Aµ +2, +Bµ +2 = 6 +√ +3Aµ +3, +Bµ +3 = ˚ +O1 +αβγ˜ϵµαβγ, +(4.54a) +Aµ +4 = 1 +3 +� +Bµ +1 + 1 +√ +3Bµ +2 +� +, +ˇEµν = ˆEµν + B3 +(µSν) − 1 +2gµνSαBα +3 , +(4.54b) +which tell us how the functional derivatives in the two frameworks are related. These do +once again suffice to prove the equivalence between the field equations in the AF and in +this framework which, for the sake of clarity and in lack of a better name, we call d-AF◦, +with the d reminding us that we use the dual of Hλµν. +5 +Exciting the connection: a series of examples +It has been standard practice in the MAG community to motivate actions from a geometric +perspective and with full generality in mind. Although this is in general good practice, +it often leads to an intractable set of field equations; in the end one will unavoidably +sacrifice generality to get results. In this section we propose an different motivation for +MAG models. Using the alternative frameworks we presented, we write meaningful field +theories propagating some of the new field variables. These are, in essence, MAG theories +propagating certain connection DoF in a tractable and controllable manner. +These MAG theories are inspired by Riemannian theories with additional fields. In +fact, they yield an effective7 set of field equations which is very much identical to the +corresponding system of equations in the Riemannian case, the crucial difference being +that instead of additional fields we use specific modes of the distortion tensor. The obvious +7The word “effective” is used to denote that one has eliminated auxiliary field variables, i.e., fields which +do not appear with time derivatives in the field equations. +– 16 – + +advantage is that for a given symmetry ansatz, we exactly know how to solve the differential +equations that arise. Hence, one should not expect metric solutions novel in form; the only +novelty is that these known metric backgrounds are now part of a larger solution with +torsion and non-metricity. In what follows, we will occasionally omit space-time indices (or +internal indices) when they are trivially understood. +5.1 +The MAGswell theory +The action for a Maxwell field Aµ coupled to four-dimensional gravity with a cosmological +constant is +˜IEM = 1 +2 +� √−gd4x +� +˜R − 2Λ − 1 +2F 2 +� +, +(5.1) +where Fµν = 2∂[µAν] is the field-strength tensor. The above integral is invariant under +shifts A′ = A + dθ, where θ(x) is some scalar potential. Variation with respect to the +metric yields the metric field equations +˜Gµν + Λgµν = FµαFνα − 1 +4F 2gµν, +(5.2) +where ˜Gµν = ˜Rµν − 1 +2 ˜Rgµν is the Einstein tensor. The Maxwell equations and the Bianchi +identity can be written as +∂ν +�√−gF νµ� += 0, +∂ν +�√−g ∗ F νµ� += 0, +(5.3) +respectively, where ∗F µν = 1 +2˜ϵµνρλFρλ is the Hodge dual of the field-strength tensor. +Here, we wish to write down a more or less similar theory for a massless vector in +MAG, the homophonous MAGswell field, Cµ, as we may playfully dub it. We will do so +by considering the Ricci scalar as our cornerstone and adding proper terms to it. Since the +HP action is invariant under projective transformations we would like to retain this feature +in the complete action. This will also allow us to work in the AF◦. We emphasize that the +MAGswell field should be understood as part of the geometry of space-time, i.e., it is not +the familiar gauge connection. +Without further ado, let us consider a fairly general projective-invariant candidate for +the four-dimensional MAGswell action which in the d-AF◦ assumes the form +IC = +� √−gd4xLC ≡ +� √−gd4x (LHP + Lct + Lkin) , +(5.4) +with +LHP = 1 +2 (R − 2Λ) , +(5.5a) +Lct = −3 +4gµνBA +µ BB +ν ηAB, +(5.5b) +Lkin = −1 +4F 2 +(C). +(5.5c) +Here, F (C) +µν += 2∂[µCν] with Cµ ≡ αABA +µ being the composite MAGswell field and αA +dimensionless constants, i.e., real non-zero numbers. The curvature scalar R stands for +˜R + ˆRT + ˆRV + 3 ˜∇µBµ +1 +(5.6) +– 17 – + +with the constituents given in (4.42). The form of the action in the AF, or the FF, can be +easily obtained by remembering that +B1 +µ = 1 +3 (2Tµ + uµ) = 1 +3 +� ˇQµ − Qµ + 2Tµ +� +, +(5.7a) +B2 +µ = +1 +6 +√ +3 (ρµ + 2uµ) = +1 +6 +√ +3 +� +4 ˇQµ − Qµ +� +. +(5.7b) +For example, in the FF we have +Lct = − 1 +36 +ˇQµ ˇQµ − 1 +9 +ˇQµQµ + 11 +144QµQµ + 1 +3 +� ˇQµTµ − QµTµ + TµT µ� +, +(5.8) +and Lkin is a linear combination of F 2 +(T), F 2 +(Q), F 2 +( ˇQ), F (T) +µν F µν +(Q), F (T) +µν F µν +( ˇQ) and F (Q) +µν F µν +( ˇQ), +where +F (T) +µν = 2∂[µTν], +F (Q) +µν += 2∂[µQν], +F ( ˇQ) +µν += 2∂[µ ˇQν]. +(5.9) +We remind the reader that we work in natural units with the reduced Planck mass further +set to unity. A quick inspection of the action indicates that the role of Lct is that of a +counter-term Lagrangian introduced to eradicate the mass terms for the BA +µ ’s, entering via +the Ricci scalar, in order for Cµ to appear massless. +Symmetries in the AF◦ (or d-AF◦). +The action (5.4) is invariant under the simulta- +neous shifts +B′A = BA + bA +(5.10) +with +b2 +µ = ∂µθ − α1b1 +µ +α2 +, +(5.11) +where θ(x) is some real scalar potential. The number of free gauge parameters to be fixed +is thus two. +Symmetries in the AF. +The action (5.4) in the AF is invariant under the simultaneous +shifts +A′I = AI + bI−1 +(5.12) +with +b3 +µ = −12α1b1 +µ + α2 +√ +3b2 +µ − 18∂µθ +6α1 + 2 +√ +3α2 +(5.13) +The number of free gauge parameters to be fixed is three. +Symmetries in the FF. +The action (5.4) in the FF is invariant under the connection +transformation +Γ′λ +µν = Γλ +µν + aλgµν + δλ +ν bµ + δλ +µcν, +(5.14) +with +bµ = (α1 + α2 +√ +3)aµ − ∂µθ +α1 − α2 +√ +3 +. +(5.15) +The number of free gauge parameters to be fixed is three. +– 18 – + +Indeed, if #fgp(−) outputs the number of free gauge parameters in a certain framework, +then #fgp(FF) = #fgp(AF) in all cases. The fact that #fgp(AF◦) = #fgp(AF) − 1 has to do +with the projective-symmetry “charge” being initially absorbed into the field variables of +the AF◦. Having discussed the symmetries in the different frameworks, we now turn our +attention to the field equations. +In the d-AF◦, they read +˚ +Oλµν +2 +≡ 1 +2 +� +˚tλµν − ˚q[µν]λ� += 0, +˚ +Oλµν +3 +≡ −1 +4˚πλµν = 0, +˚ +Oλµν +4 +≡ ˚ +O(µν)λ +2 ++ 1 +8˚qλµν = 0, +(5.16a) +Bµ +1 ≡ α1 +α2 +Bµ +2 ≡ α1 ˜∇νF νµ +(C) = 0, +Bµ +3 ≡ 3 +2Sµ = 0, +(5.16b) +2 ˇEµν ≡ ˜Gµν − 1 +2gµν +� +ˆRT − 2Λ + 3 +2S2 − 1 +2F 2 +(C) +� +− 3 +4˚πµαβ˚πναβ + ++˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 +2 +� +˚qµαβ˚qναβ +˚tµαβ˚tναβ� ++ ++˚tαβ(µ˚qν) +αβ − ˚qαβ +(µ˚tν)αβ −˚tβα(µ˚qαβ +ν) + 3 +2SµSν − F (C) +µ +αF (C) +να . +(5.16c) +The first three of them imply ˚ +OI +λµν = 0. Then, equation Bµ +3 = 0 suggests that Sµ = 0 +which yields Hλµν = 0 via (2.4). Substituting these results back into (5.16), we obtain the +effective set +∂ν +�√−gF νµ +(C) +� += 0 = ∂ν +�√−g ∗ F νµ +(C) +� +, +(5.17a) +˜Gµν + Λgµν = F (C) +µ +αF (C) +να − 1 +4F 2 +(C)gµν, +(5.17b) +where we took the liberty to also include the Bianchi identity for F (C) +µν +with +∗ F µν +(C) = 1 +2˜ϵµνρλF (C) +ρλ . +(5.18) +As differential equations, these exactly correspond to the Einstein-Maxwell system. Let us +now study the solution in the different frameworks. +Solution in the d-AF◦. +As already mentioned, we have ˚ +OI +λµν = 0 = Sµ and eq. (5.17a), +which only determines the combination Cµ. If ⟨Cµ⟩ + ∂µφ is the value of Cµ ≡ αABA +µ +satisfying (5.17a), then BA acquires the value ⟨BA⟩ with +⟨B2 +µ⟩ = ⟨Cµ⟩ + ∂µφ − α1⟨B1 +µ⟩ +α2 +. +(5.19) +Clearly, the values ⟨BA⟩ + bA, where the bA’s obey eq. (5.11), are also acceptable. A good +strategy to capture the solution in all possible gauges is to set +b1 = −⟨B1⟩ + ˜α⟨C⟩, +θ = −φ, +(5.20) +– 19 – + +where ˜α is a real number, obtaining +B1 = ˜α⟨C⟩, +B2 = 1 − ˜αα1 +α2 +⟨C⟩. +(5.21) +Therefore, +{B1, B2} = +� +� +� +� +� +� +� +{0, ⟨C⟩/α2} +˜α = 0 +{⟨C⟩/α1, 0} +˜α = 1/α1 +{˜α⟨C⟩, (1 − ˜αα1)⟨C⟩/α2} +˜α ̸= 0, 1/α1. +(5.22) +Solution in the AF. +The next step is to translate the solution into the language of the +AF. The field equations tell us that ˚ +ON +λµν = 0, and that eq. (5.17a) must hold true. Again, +if ⟨Cµ⟩ + ∂µφ is the value of +Cµ ≡ α2 +6 +√ +3ρµ + 2α1 +3 Tµ + 3α1 + +√ +3α2 +9 +uµ +(5.23) +satisfying (5.17a), then AI acquires the value ⟨AI⟩ with +⟨uµ⟩ = 18 (⟨Cµ⟩ + ∂µφ) − +√ +3α2⟨ρµ⟩ − 12α1⟨Tµ⟩ +2(3α1 + α2 +√ +3) +. +(5.24) +Clearly, the values ⟨AI⟩ + bI−1, where the bI−1’s obey eq. (5.13), are as good. Setting +b1 = α⟨C⟩ − ⟨T⟩, +b2 = β⟨C⟩ − ⟨ρ⟩, +θ = −φ, +(5.25) +where α, β are real numbers, we get +T = α⟨C⟩, +ρ = β⟨C⟩, +u = 18 − 12αα1 − βα2 +√ +3 +2(3α1 + α2 +√ +3) +⟨C⟩. +(5.26) +We can further identify +˜α = 18 + +√ +3(4α − β)α2 +6(3α1 + α2 +√ +3) +, +(5.27) +and we collect the various cases in table 1. +Solution in the FF. +The final step is to translate the solution into the language of the +familiar Palatini formalism, i.e., to present an affine connection which solves the connection +field equations. This reads +Γλ +µν = ˜Γλ +µν + ⟨V λ⟩gµν − δλ +ν +⟨Cµ⟩ + ∂µφ − +� +α1 + α2 +√ +3 +� +⟨Vµ⟩ +α1 − α2 +√ +3 ++ δλ +µ⟨Uν⟩, +(5.28) +where ⟨Cµ⟩ + ∂µφ is the value of +Cµ ≡ 3α1 + 2 +√ +3α2 +9 +ˇQµ − 6α1 + +√ +3α2 +18 +Qµ + 2α1 +3 Tµ +(5.29) +– 20 – + +α +β +˜α +Tµ +ρµ +uµ +0 +0 +3 +3α1 +α2 +√ +3 +0 +0 +✓ +0 +6 +√ +3 +α2 +0 +0 +✓ +0 +3 +2α1 +0 +1 +α1 +✓ +0 +0 +̸= 0, +3 +2α1 +2 +√ +3(3−2αα1 ) +α2 +2α +3 +✓ +✓ +0 +̸= 0, +3 +2α1 +0 +9+2 +√ +3αα2 +9α1 +3α2 +√ +3 +✓ +0 +✓ +0 +̸= 0, 6 +√ +3 +α2 +18− +√ +3βα2 +6(3α1 +α2 +√ +3) +0 +✓ +✓ +̸= 0 +̸= 0, 2 +√ +3(3−2αα1 ) +α2 +18+ +√ +3(4α−β)α2 +6(3α1 +α2 +√ +3) +✓ +✓ +✓ +Table 1. +Values of ˜α and the AF field variables AI for different numbers α, β. A checkmark ✓ +indicates that the ticked field is proportional to ⟨Cµ⟩. +satisfying (5.17a). +Again, due to the freedom to shift our connection as in (5.14), and +setting +a = 24 − (4α − β)(α1 − α2 +√ +3) +12(3α1 + α2 +√ +3) +⟨C⟩ − ⟨V ⟩, +θ = −φ, +(5.30a) +c = (8α + β)α2 +1 − 4α1(3 + 2αα2 +√ +3) − 3α2(βα2 − 4 +√ +3) +12(3α2 +1 − 2α1α2 +√ +3 − 3α2 +2) +⟨C⟩ − ⟨U⟩, +(5.30b) +we reach a connection with torsion and non-metricity +T λµν = 2α +3 δλ +[ν⟨Cµ]⟩, +(5.31a) +Qλµν = β +6 g(λµ⟨Cν)⟩ + 18 − 12αα1 − βα2 +√ +3 +9(3α1 + α2 +√ +3) +� +gλ(µ⟨Cν)⟩ − gµν⟨Cλ⟩ +� +. +(5.31b) +One can immediately verify that if we decompose the latter under the Lorentz group, we +will exactly find (5.26) as the only excited irreducible modes. Therefore, one can again refer +to table 1 for the various cases. +An interesting remark is in order. +The Lagrangian does undeniably propagate the +massless combination Cµ, a spin-1 geometric “boson”. This means that part of the post- +Riemannian structure gets (self-)excited but it turns out to be impossible to make a gauge- +independent statement about specifically which part that is. For example, what appears +to be an excitation of only torsional DoF in one gauge, shows up as an excitation of only +non-metricity DoF in another. Hence, propagation of the MAGswell field is tantamount +to a self-excitation of the connection background with different parts of the latter being +excited in the different gauges. +Do also note that the action (5.4) can be thought of as the massless limit of a massive +theory which has an action like (5.4), but with Lct replaced by +Lmass = −1 +2 +� +(µ2α2 +1 − 3)B1 +µB1 +ν + 2µ2α1α2B1 +µB2 +ν + (µ2α2 +2 + 3)B2 +µB2 +ν +� +gµν, +(5.32) +– 21 – + +such that, up to surface terms, +IC = 1 +2 +� √−gd4x +� +˜R − 2Λ + ˆRT + 3S2 − µ2C2 − 1 +2F 2 +(C) +� +, +(5.33) +always in the d-AF◦. Obviously, Lct = Lmass(µ = 0). The last two terms in (5.33) imply +that the combination Cµ behaves as a Proca field with mass µ. Since the HP action already +introduces a mass scale proportional to the Planck mass,8 naturalness criteria suggest that +we take µ to be of the same order (the composite field Cµ is part of the space-time geometry, +not some external field). The field equations in the d-AF◦ are (5.16) except that ˜∇νF νµ +(C) = 0 +is replaced by the Proca equation +˜∇νF νµ +(C) − µ2Cµ = 0. +(5.34) +Observe that the massive action and the field equations following from it, do still possess +a symmetry under (5.10) if θA = 0. This means that the propagated combination Cµ is a +massive vector-boson, and the geometric interpretation of this propagation again falls into +the previous scheme, viz, it is subject to the choice of gauge. +Finally, we remark that the MAGswell field is of course by itself not a solid and unique +concept. +Nevertheless, let us justify why we think that Cµ is indeed the most general +candidate to describe it. First of all, playing the devil’s advocate, one could argue that +there are more general projective-invariant combinations to take as our BA fields; we have +already shown this in section 4. Indeed, there is simply no physical argument favoring (4.21) +over (4.26). Sure, the diagonal form of the mass-squared matrix and the emergence of SO +symmetries under transformations in field space are nice features, but they are far from +being necessary restrictions. Actually, these features are completely absent here because (i) +we have removed the mass terms, and (ii) there are no SO transformations being a symmetry +of (5.4). +However, the real question is if we would gain more insight by considering a +more complicated change of field variables. The answer is no, for Cµ would again be a +linear combination of the AI’s but with different coefficients. Assuming that we would also +properly modify Lct as to remove algebraic instances of the new BA’s, it is evident that we +would not get qualitatively different results. +Second, one could argue that projective invariance of the action is definitely not manda- +tory. One could instead add a kinetic term for any of the vector variables in the AF and +remove all algebraic instances of this field from the action by introducing a proper counter- +term Lagrangian. This theory, which would no longer be invariant under (4.11) (the AF +analogue of what is a projective transformation in the FF), would then propagate the gravi- +ton and the specific mode. Fortunately, one can easily prove that the solutions in all these +different cases would correspond to the one and only solution in the theory with action (5.4) +in different gauges. +5.2 +Non-linear interacting MAG theory, black holes and solitons +To showcase the usefulness of this new approach to MAG for obtaining exact solutions +with torsion and non-metricity, let us propose a four-dimensional interacting action with +8Remember that we have set the reduced Planck mass to unity. +– 22 – + +non-linear dynamics for the MAGswell field Cµ and the pseudo-vector Sµ, namely +INL = +� √−gd4x +�1 +2(R − 2Λ − ˆRV ) − γ1F 2 +(C) − γ2F 2 +(S) − γLint +� +, +(5.35) +where +Lint = δµ1...µ4 +ν1...ν4 F (C) +µ1µ2F (S) +µ3µ4F ν1ν2 +(C) F ν3ν4 +(S) , +(5.36) +γ is a positive coupling constant of mass-dimension −4, and γ1, γ2 are positive coupling +constants of mass-dimension 0. This is the form of the action in the d-AF◦. The field +strengths can be customarily written using only the partial derivative, i.e., F(C) as previously +defined and F (S) +µν += ∂[µSν] with Sµ given in eq. (2.4). The term R stands for ˜R + ˆRT + +ˆRV + t.d., with the constituents defined in eqs. (4.42). One can always express the action +in the FF (or the AF) by recalling eqs. (5.7). This interacting Lagrangian was proposed +in [40] for two distinct potentials in a Riemannian setup. Here, we endow these potentials +with a special geometric origin and cast the whole thing as a MAG theory. +In the d-AF◦, the field equations read +˚ +Oλµν +2 +≡ 1 +2 +� +˚tλµν − ˚q[µν]λ� += 0, +˚ +Oλµν +3 +≡ −1 +4˚πλµν = 0, +˚ +Oλµν +4 +≡ ˚ +O(µν)λ +2 ++ 1 +8˚qλµν = 0, +(5.37a) +Bµ +1 +4α1 +≡ Bµ +2 +4α2 +≡ γ1 ˜∇νF νµ +(C) − γδµνρσ +αβγδF (S) +ρσ ˜∇ν +� +F αβ +(C)F γδ +(S) +� += 0, +(5.37b) +Bµ +3 +4 +≡ γ2 ˜∇νF νµ +(S) + γδνρσµ +αβγδF (C) +νρ ˜∇σ +� +F αβ +(C)F γδ +(S) +� += 0, +(5.37c) +2 ˇEµν ≡ ˜Gµν − 1 +2gµν +� +ˆRT − 2Λ − 2γ1F 2 +(C) − 2γ2F 2 +(S) + 2γLint +� ++ ++˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 +2 +� +˚qµαβ˚qναβ +˚tµαβ˚tναβ� ++ ++˚tαβ(µ˚qν) +αβ − ˚qαβ +(µ˚tν)αβ −˚tβα(µ˚qαβ +ν) − 3 +4˚πµαβ˚πναβ − +−4γ1F (C) +µ +αF (C) +να − 4γ2F (S) +µ +αF (S) +να = 0. +(5.37d) +To get the above expressions, we also used the Bianchi identities dF(C) = 0 = dF(S) and +the dimension-dependent identity +δµ1...µ4 +ν1...ν4 F (C) +[µ1µ2F (S) +µ3µ4F ν1ν2 +(C) F ν3ν4 +(S) gµ]ν = 0. +(5.38) +The action and the field equations are invariant under the transformation (5.10) with +the bA’s constrained via (5.11). They are also invariant under a shift of Sµ by a locally +exact co-vector, i.e., S′ +µ = Sµ + ∂µφ, which in the AF amounts to +H′ +λµν = Hλµν − ˜ϵλµνα∂αφ, +(5.39) +whereas it corresponds to the connection transformation +Γ′λ +µν = Γλ +µν + 1 +2˜ϵλµνα∂αφ, +(5.40) +– 23 – + +in the FF. Following the steps laid down in the previous section, we expect a connection +solution with torsion and non-metricity +T λµν = 2α +3 δλ +[ν⟨Cµ]⟩ + ⟨Sα⟩˜ϵαλµν, +(5.41a) +Qλµν = β +6 g(λµ⟨Cν)⟩ + 18 − 12αα1 − βα2 +√ +3 +9(3α1 + α2 +√ +3) +� +gλ(µ⟨Cν)⟩ − gµν⟨Cλ⟩ +� +, +(5.41b) +respectively, where ⟨C⟩ and ⟨S⟩ satisfy the non-linear differential equations (5.37b) and (5.37c). +We remind the reader that our gauge freedom is fully exhausted once we fix values for α, β +(see table 1). +Now, let us consider the static spherically-symmetric metric ansatz +ds2 = −f(r)dt2 + dr2 +f(r) + r2dΣ2 +2, +(5.42) +where dΣ2 +2 = dχ2 + sin2 χdy2 gives the line element of a two-dimensional spherical section +with χ, y compact. We also make the following ansätze, +Cµ = c(r)δ0 +µ, +Sµ = p cos χδ3 +µ, +(5.43) +which result in +F (C) +µν += c′δ10 +µν, +F (S) +µν = p sin χδ32 +µν. +(5.44) +A prime denotes differentiation with respect to r. +Given the above, eq. (5.37c) is identically satisfied, while eq. (5.37b) gives +r(8γp2 + γ1r4)c′′ + 2(γ1r4 − 8γp2)c′ = 0. +(5.45) +This yields the first integral +c′ = − +qr2 +γ1r4 + 8γp2 , +(5.46) +where q is an integration constant. Integrating once more, we get +c = c0 + +q +γ1r 2F1 +�1 +4, 1, 5 +4; −8γp2 +γ1r4 +� +, +(5.47) +where 2F1 is the Gaussian hypergeometric function [41], and c0 is another constant of +integration. Therefore our connection solution is such that its torsion and non-metricity +read +T λµν = −α +3 δλ0 +µνc + p cos χ˜ϵ3λµν, +(5.48a) +Qλµν = c +� +β +6 g(λµδ0 +ν) + 18 − 12αα1 − βα2 +√ +3 +9(3α1 + α2 +√ +3) +� +gλ(µδ0 +ν) − gµνδ0 +λ +�� +, +(5.48b) +with c given in (5.47). +Plugging this into (5.37d), we find that +−2r +f +ˇE00 = f′ + f +r − k +r + Λr + 2γ2p2 +r3 ++ +2q2r +γ1r4 + 8γp2 = 0. +(5.49) +– 24 – + +Since +ˇE11 = −f−2 ˇE00, +ˇE33 = sin2 χ ˇE22, +(5.50) +and +ˇE22 = r2 +2f +ˇE00 − +� r3 +2f +ˇE00 +�′ +, +(5.51) +we only have to find the solution to eq. (5.49), which reads +f = 1 − 2M +r +− Λr2 +3 ++ 2γ2p2 +r2 ++ 2q2 +γ1r2 2F1 +�1 +4, 1, 5 +4; −8γp2 +γ1r4 +� +. +(5.52) +The symbol M stands for yet another integration constant, this time associated with the +mass. The very interesting metric background (5.52) has been extensively studied in [40, 42], +and there is no need to discuss it here in depth. In our case, nevertheless, the corrections to +the Schwarzschild-(A)dS metric is due to a richer space-time geometry and not due to the +introduction of additional fields (like a Maxwell field). In this sense, this is a novel result. +Some comments are in order. Observe that by setting p = 0, the background (5.52) +assumes the form +f = 1 − 2M +r ++ 8q2 +r2 − Λr2 +3 , +(5.53) +and, up to choice of the integration constant q, it is indeed the metric solution in the +MAGswell theory with action (5.4) if we make the ansatz (5.43) for Cµ. Moreover, the +torsion and non-metricity of the connection solution, eqs. (5.48), acquire the form (5.31), +ergo, we recover the full solution in the MAGswell model, as a special case. +Another +interesting setup is to consider the action (5.35) with Λ = 0 = γ1. +In this case, the +connection solution will have torsion and non-metricity (5.48) with +c = c0 + qr3, +(5.54) +whereas the metric function f will be +f = 1 − 2M +r ++ 2γ2p2 +r2 +− Λeffr2 +3 +, +(5.55) +where Λeff > 0 stands for the effective cosmological constant +Λeff = 16γp2q2. +(5.56) +Finally, non-singular solutions were reported in [40] for a specific choice of the mass +parameter M in a strongly-coupled regime. The need to go to such a regime will not be +necessary here; we will just set γ2 = 0 and choose our mass parameter as +M = M∗ := +πq2 +4(2γp2γ3 +1)1/4 . +(5.57) +Then, eq. (5.52) assumes the expression +f = 1 − 2M∗ +r +− Λr2 +3 ++ 2q2 +γ1r2 2F1 +�1 +4, 1, 5 +4; −8γp2 +γ1r4 +� +, +(5.58) +– 25 – + +and admits the near-origin expansion +f = +r→0 1 − +� +q2 +12γp2 + Λ +3 +� +r2 + O(r3). +(5.59) +If Λ ≥ 0 or −q2/(4γp2) < Λ < 0, the presence of a de Sitter core with radius +ldS = +2p√3γ +� +q2 + 4γp2Λ +, +(5.60) +is manifest, ensuring regularity of Riemann-curvature invariants at the origin and com- +pleteness in the geodesic sense [43]. A further study of the causal structure of the solution +reveals [40] that, for certain values (or ranges thereof) of the coupling/integration constants, +eq. (5.58) describes either a gravitational soliton (horizon-free solution with regular origin), +or just a standard black hole solution with an extremal limit. To the best of our knowledge, +regular black hole solutions with torsion and non-metricity have not been yet reported in +the MAG literature. +Since the actual novelty in the full solution is the existence of a non-trivial connection +background, we find it worth to include a few lines about the behavior of the latter in +various limits. First, let us write torsion and non-metricity in a coordinate-free manner by +introducing a vierbein field eµ +a, with indices a, b, ... = (0), ..., (3) and inverse ea +µ satisfying +the orthonormality relation gµν = ηabea +µeb +ν. In particular, let us choose it to be diagonal, +viz., +ea +µ = diag +�� +f, 1 +√f , r, r sin χ +� +. +(5.61) +Then, the only non-vanishing components of T abc = ea +λeµ +b eν +cT λµν are +T (0) +(1)(2) = T (1) +(0)(2) = −T (2) +(0)(1) = p cot χ +r +, +T (i) +(0)(i) = +αc +3√f , +(5.62) +where i, j, ... take values in {1, 2, 3}. It seems that the (i)(0)(i) components will be singular +at the horizon radius r = r+. Fortunately, this can be remedied by fixing the integration +constants c0 in (5.47) as +c0 = − +q +γ1r+ +2F1 +�1 +4, 1, 5 +4; −8γp2 +γ1r4 ++ +� +, +(5.63) +so that c ∼ (r − r+) near the horizon surface. Moreover, all components of the torsion +tensor exhibit a +r−1 fall-off at asymptotic infinity. Next, we have a single pole at the +origin r = 0 due to the axial part. This pole persists even in the case of the regular metric +solution (5.58). If we assume that a probe particle with micro-structure follows the auto- +parallels, then it is a good question to ask whether this particle is going to “feel” the torsion +singularity at the origin. Thankfully, the axial part of torsion drops out of the auto-parallel +equation [20], and thus, this singular behavior should not really be a cause for concern! +Finally, all components of Qabc = eλ +aeµ +b eν +cQλµν are proportional to c/√f. For f as in (5.52), +this ratio vanishes at all previously discussed radii. On the other hand, in the case of the +regular metric (5.58), it acquires a finite value at the origin. In the regular extremal case, +it further is finite also at r = r+. +– 26 – + +5.3 +Cosmological constant powered by torsion +It is an old fact that the minimal coupling of a 3-form field to Einstein gravity without +a cosmological constant leads to Einstein’s field equations with a cosmological constant +purely derived from a gauge principle [44]. Here, we shall disclose a MAG model with no +cosmological constant which also leads to pure gravity with a cosmological constant, the +latter now powered by axial torsion. +Let us consider the projective-invariant action +IH = 1 +2 +� √−gd4x +� +R + 1 +4H2 − 1 +24F 2 +(H) +� +, +(5.64) +where F (H) +λµνρ = 4∂[λHµνρ]. The purpose of the second term in the above integral is to cancel +out the mass term for Hλµν present in the AF expression of the Ricci scalar. This ensures +that the action (5.64) is invariant under the translation +H′λµν = Hλµν + ∂[λBµν] +(5.65) +which corresponds to the transformation +Γ′λ +µν = Γλ +µν − 1 +2gλρ∂[ρBµν] +(5.66) +of the affine connection in the FF, with Bµν being an arbitrary 2-form field. +In the convenient AF◦, the field equations read +˚ +Oλµν1 = 1 +6 +˜∇αF αλµν +(H) += 0, +(5.67a) +˚ +Oλµν +2 +≡ 1 +2 +� +˚tλµν − ˚q[µν]λ� += 0, +˚ +Oλµν +3 +≡ −1 +4˚πλµν = 0, +˚ +Oλµν +4 +≡ ˚ +O(µν)λ +2 ++ 1 +8˚qλµν = 0, +(5.67b) +Bµ +1 ≡ −3 +2Bµ +1 = 0, +Bµ +2 ≡ 3 +2Bµ +2 = 0, +(5.67c) +2 ˆEµν = ˜Gµν − 1 +2gµν +� +RT + RV + 1 +4H2 − 1 +24F 2 +(H) +� +− 3 +4˚πµαβ˚πναβ + ++˚qαβµ˚qαβν + 1 +2˚qµαβ˚qναβ +˚tαβµ˚tαβν + 1 +2 +˚tµαβ˚tναβ +˚tαβ(µ˚qν) +αβ − +−˚qαβ +(µ˚tν)αβ −˚tβα(µ˚qαβ +ν) + 3 +2BA +µ BB +ν ηAB − 1 +6F (H) +µ +αβγF (H) +ναβγ = 0 +(5.67d) +where the expressions of the BA +µ ’s in terms of the AF (or FF) variables are to be found in +eqs. (5.7), RT in (4.9) (first one), and RV in (4.22). From the above, it is quite easy to +conclude that ˚ +OI = 0 = BA. Therefore, the field equations assume the effective form +˜∇αF αλµν +(H) += 0, +(5.68a) +˜Gµν = 1 +48 +� +8F (H) +µ +αβγF (H) +ναβγ − gµνF 2 +(H) +� +. +(5.68b) +To proceed, one must now recall that +F (H) +λρµν = χ(x)˜ϵλρµν, +(5.69) +– 27 – + +since F(H) is a top-form in four dimensions. Clearly, equation (5.68a) implies that χ is an +integration constant, say equal to χ0. Consequently, we are left with +˜Gµν + 1 +2χ2 +0gµν = 0, +(5.70) +which will determine the metric, and we directly find that +˜Rµν = χ2 +0 +2 gµν, +(5.71) +which is the familiar Riemannian Ricci-curvature condition for Einstein manifolds with +positive constant curvature. +As promised, we found a connection solution which features only axial torsion Hλµν. +Again, we stress out that this type of torsion has no effect on the auto-parallels, i.e., the +latter continue to coincide with the geodesics. We also saw that our field equations do ef- +fectively become Einstein’s field equations with a positive (effective) cosmological constant, +Λeff = χ2 +0/2, once we integrate out the connection. The cosmological solution in the ab- +sence of matter sources would then be a de Sitter universe with Hubble constant H ∝ |χ0| +where the expansion is now driven by an actual integration constant, powered by torsion, +instead of an a priori fixed value. +6 +Summary and future prospects +We started from the observation that the affine connection is a single field encoding n3-many +off-shell DoF, arguing that, for certain purposes, it might be more efficient to distribute +these degrees among more than one fields. We then proceeded with a convenient change of +field variables {g, Γ} → {g, ˚ +ON, AI} going to a framework which we dubbed AF. Besides +the metric, the new field variables are the irreducible pieces of the torsion and non-metricity +tensors under the Lorentz group. They are thus automatically identified with the funda- +mental fields {g, Γ} in the FF. +We worked out in detail the relations between the functional derivatives in the two +frameworks and concluded that, not surprisingly, the field equations in the AF imply and +are implied by the field equations in the FF. Hence, the field equations in the AF constitute +an equivalent system, and we have the freedom, at any stage, to switch between the different +frameworks. To complete the mapping, we further disclosed a correspondence between linear +connection transformations in the FF and translations in the AF while we also determined +how the ˚ +ON’s and the AI’s should transform under a local Weyl re-scaling of the metric. +We then applied the AF to the Hilbert-Palatini action and showed its well-known equiv- +alence to Einstein gravity (up to choice of a gauge) also in the new framework. Observing +that a projective transformation of the connection corresponds to simultaneous translations +of the AI’s in the AF, we further argued that the projective symmetry manifests itself as a +true gauge symmetry in the new framework, i.e., one of the components of the vector triplet +Aµ is redundant. In particular, this means that any projective-invariant action admits a +description in terms of a reduced set of variables {g, ˚ +ON, BA} where the BA’s are in general +– 28 – + +identified with linear combinations of the AI’s. This led us to develop a useful variant of +the AF, which we dubbed diminished AF or AF◦ for short. +We saw that there exists a particular choice of combinations BA which reveals an +SO(1, 1) symmetry of the n-dimensional HP action under a group action on the components +of the doublet Bµ. In n = 4, the field variables in the AF◦ can be re-organized. Using the +fact that the dual of the 3-form torsion is a pseudo-vector, the quadruplet ˚ +Oλµν is reduced +to a triplet by handing its first component to the doublet Bµ which becomes a triplet. This +is just a special four-dimensional variant of the AF◦, obtained via the change of variables +{g, ˚ +ON, BA} → {g, ˚ +OI, BA}, which we called d-AF◦ for the sake of clarity. As it turns +out, the HP action proves to be an SO(1, 2)-symmetric action in the d-AF◦ where the +group action mixes the BA’s. +Actions of the SO(2) subgroup rotate the elements of a +two-dimensional subspace with the discrete version for θ = π/2 interpreted as a rotation of +axial torsion to non-metricity, and vice versa. +Observing that any MAG theory in these alternative frameworks can be handled as a +Riemannian theory with additional fields, we argued that it is an efficient strategy to use +solvable (and suitable) Riemannian theories as “seeds” for solvable MAG theories which +propagate the connection in vacuum. As our first example, we drew inspiration from the +elegant Einstein-Maxwell theory. We proposed a theory for what we called the MAGswell +field, a composite field labeling a projective-invariant linear combination of torsion and +non-metricity traces. The naive action should follow from the Maxwell action by replacing +˜R with R and the gauge field with the MAGswell field, the latter having nothing to do with +a gauge connection. Doing so, one of course notices that what was a U(1) of the second +kind in the Riemannian case does not translate into a symmetry of the MAG theory under +locally exact shifts of the MAGswell field. The reason is that the presence of the Ricci scalar +makes the field massive. Thus, a counter-term Lagrangian was also included with the sole +purpose of removing the mass terms for the constituents of the composite MAGswell field. +We then discussed the symmetries of the MAGswell action in all frameworks. Exactly +because the MAGswell field is a composite object, we showed that the action is symmetric +under a 2-parameter transformation of the vector variables in the AF◦ (or the d-AF◦), +which combines a transformation preserving the MAGswell field and one translating it by +an exact vector. In the other frameworks, this symmetry shows up as a symmetry under a +3-parameter transformation, a fact attributed to the absorption of the projective-symmetry +charge in the diminished AF. We derived the field equations in the d-AF◦ and presented the +solution in all frameworks, finding a proper expression that captures its form in all gauges. +Actually, the reader was provided with table 1 which displays all cases possible, and which +proves that the propagation of the MAGswell field, a gauge-independent fact, cannot be +tied to a self-excitation of a uniquely determined part of the post-Riemannian structure in +a gauge-independent fashion, i.e., different parts of the connection background get excited +for different choices of gauge. +After this instructive example, we proceeded with a more complicated theory, this time +inspired by quasi-topological electromagnetism [40]. We proposed a Lagrangian with non- +linear dynamics for the MAGswell field and the torsion pseudo-vector letting them interact +with each other. After briefly discussing the symmetries and deriving the field equations, +– 29 – + +we adopted a static and spherically-symmetric metric ansatz, together with compatible +ansätze for torsion and non-metricity, in an attempt to recover the black hole solution +reported in [40, 42]. The full solution describes a black hole with a non-zero connection +background sourcing the post-Schwarzschild contributions to the metric solution. Under a +certain tuning of the integration constants, we also showed that this black hole exhibits a +regular core and is thus complete in the geodesic sense. +However, assuming that particles with micro-structure follow auto-parallels, we also +had to analyze the behavior of the torsion and non-metricity of the solution at all radii +of interest. Doing so, we had to fix yet another integration constant to avoid a singular +behavior at the horizon radius, but we concluded that there is no remedy for a single pole at +the origin due to axial torsion. This pole is inevitable even in the case of the regular black +hole. Nevertheless, as we pointed out, the axial piece of torsion drops out from the auto- +parallel equation meaning that the probe particle would never be affected by this torsion +singularity. +Finally, as our last example, and inspired by the derivation of a cosmological constant +from a gauge principle [44], we put forth a simple MAG action for the 3-form torsion. After +deriving the field equations we presented a connection solution featuring only axial torsion +which powers a positive effective cosmological constant. The cosmological solution in this +MAG theory — in the absence of matter sources — would be a de Sitter universe with the +expansion driven by torsion. We remarked that the effective cosmological constant is an +integration constant as opposed to a fixed-value Λ introduced by hand in the action. +The main goal of this work was to communicate the idea that a smart change of field +variables can be a really useful strategy when trying to find solvable MAG theories. Indeed, +our proposal proves to be a fruitful one, for although we restricted ourselves to showing +only three examples, these are suggestive of many more. Writing down simple field theories +for the new field variables compared to considering combinations of curvature invariants to +make the connection dynamical, is of course a far less general method, albeit a much more +targeted and result-oriented one. In the future, we plan to give more examples and solutions +which are not necessarily inspired by Riemannian theories. We find it interesting to study +kinetic theories for the various tensor modes and also investigate if (and how) Riemannian +theories with scalar fields can fit as an inspiration into this description of MAG. +A +Irreducible decomposition of a rank-3 tensor +The irreducible decomposition of a general rank-3 tensor ∆λµν under the Lorentz group +reads +∆λµν = ∆[λµν] + ˚∆(λµν) + ˚ +Dλ[µν] + ˚ +Dλ(µν) + ¯∆(λµν) + ¯Dλ[µν] + ¯Dλ(µν), +(A.1) +– 30 – + +where +˚ +Dλµν = ∆λµν − ∆(λµν) − ∆[λµν] − +1 +n − 1gλ[µ +� +∆α +|α|ν] − ∆α +ν]α +� +− +− +1 +3(n − 1)gλ(µ +� +∆α +|α|ν) + ∆α +ν)α − 2∆ν)α +α� ++ ++ +1 +3(n − 1)gµν (∆ααλ + ∆αλα − 2∆λαα) , +(A.2a) +˚∆(λµν) = ∆(λµν) − +1 +D + 2g(µν +� +∆λ)α +α + ∆α +λ)α + ∆α +|α|λ) +� +, +(A.2b) +¯∆(λµν) = +1 +n + 2g(µν +� +∆λ)α +α + ∆α +λ)α + ∆α +|α|λ) +� += ∆(λµν) − ˚∆(λµν), +(A.2c) +¯Dλµν = +1 +n − 1gλ[µ +� +∆α +|α|ν] − ∆α +ν]α +� +− +1 +3(n − 1)gµν (∆ααλ + ∆αλα − 2∆λαα) + ++ +1 +3(n − 1)gλ(µ +� +∆α +|α|ν) + ∆α +ν)α − 2∆ν)α +α� += Dλµν − ˚ +Dλµν. +(A.2d) +B +Glossary +Indices +Values +µ, ν, ... +0,1,...,n − 1 +i, j, ... +1,2,...,n − 1 +a, b, ... +(0),(1),...,(n − 1) +M, N, ... +1, 2, 3, 4 +I, J, ... +2, 3, 4 +A, B, ... +1, 2 +A, B, ... +1, 2, 3 +Table 2. +Indices used in this work and their values. +Acronym +Full name +MAG +Metric-affine gravity +DoF +Degrees of freedom +FF +Fundamental framework, {g, Γ} +AF +Alternative framework, {g, ˚ +ON, AI} +AF◦ +Diminished alternative framework, {g, ˚ +ON, BA} +d-AF◦ +-, {g, ˚ +OI, BA} +Table 3. +Acronyms used in this work and their full name. +– 31 – + +Symbol +Definition +˚ +Oλµν +{Hλµν,˚tλµν,˚πλµν,˚qλµν} +Aµ +{Tµ, ρµ, uµ} +Bµ in the AF◦ +(4.21) +Bµ in the d-AF◦ +(4.43) +g +det(gµν) +˚ +Oλµν +N +√−g−1δI/δ˚ +ON +λµν +Aµ +I +√−g−1δI/δAI +µ +Bµ +A/A +√−g−1δI/δBA/A +µ +∆λµν +√−g−1δI/δΓλ +µν +Eµν +√−g−1δI/δgµν in the FF +ˆEµν +√−g−1δI/δgµν in the AF +ˇEµν +√−g−1δI/δgµν in the d-AF◦ +RT , RV +(4.9) +RV +(4.22) +ˆRT , ˆRV +(4.42) +Table 4. +Some symbols used in this work and their definition. +Acknowledgments +D.I. work is funded by the Estonian Research Council grant (SJD14). 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B, 94:179–182, 1980. +– 34 – + diff --git a/wtFIT4oBgHgl3EQfzSt7/content/tmp_files/load_file.txt b/wtFIT4oBgHgl3EQfzSt7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c79b67a060d64efd6eff2194049cf704cd0207b0 --- /dev/null +++ b/wtFIT4oBgHgl3EQfzSt7/content/tmp_files/load_file.txt @@ -0,0 +1,1186 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf,len=1185 +page_content='Prepared for submission to JHEP Describing metric-affine theories anew: alternative frameworks, examples and solutions Damianos Iosifidisa,b Konstantinos Pallikaris,a,1 aLaboratory of Theoretical Physics, Institute of Physics, University of Tartu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Ostwaldi 1, 50411 Tartu, Estonia bInstitute of Theoretical Physics, Department of Physics Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece E-mail: damianos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='iosifidis@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='ee, konstantinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='pallikaris@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='ee Abstract: In this work we describe metric-affine theories anew by making a change of field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A series of equivalent frameworks is presented and identifications are worked out in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The advantage of applying the new frameworks is that any MAG theory can be handled as a Riemannian theory with additional fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We study the Hilbert-Palatini action using the new field variables and disclose interesting symmetries under SO transformations in field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Then, we use solvable and suitable Riemannian theories as seed models for solvable MAG theories, restricting ourselves to three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We present a black hole solution with torsion and non-metricity which under a certain tuning acquires a regular core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A de Sitter universe with the expansion powered by 3-form torsion, is also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11364v1 [gr-qc] 26 Jan 2023 Contents 1 Introduction 1 2 Preliminaries 3 3 The alternative framework 5 4 Revisiting the Hilbert-Palatini action 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1 FF vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' AF 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2 Projective symmetry and the AF◦ 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3 SO(1, 2) symmetry, torsion/non-metricity rotations, and the d-AF◦ 14 5 Exciting the connection: a series of examples 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1 The MAGswell theory 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2 Non-linear interacting MAG theory, black holes and solitons 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3 Cosmological constant powered by torsion 27 6 Summary and future prospects 28 A Irreducible decomposition of a rank-3 tensor 30 B Glossary 31 1 Introduction The general theory of relativity is perhaps one of the most elegantly simple theories of physics with such a strong impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The ingenious statement that matter tells space-time how to curve, and curved space-time tells matter how to move, crystallized into Einstein’s equations, Gµν = 8πG c4 Tµν, along with the many experimental tests the theory has successfully passed since its birth, have truly established it as the most widely accepted theory of gravitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Despite its successes, Einstein’s theory has its shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' To mention but a few of them, first, general relativity is not a perturbatively renormalizable quantum field theory meaning that it gets striped of its predictive power at high energies with the Planck mass constituting the cut-off scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Second, the small measured value of the cosmological constant leads to a perplexing discrepancy between theory and experiment, a naturalness problem known as the cosmological constant problem (see [1] for an extensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Prominent shortcomings are also the flatness and horizon problems [2–5] plaguing the standard model of hot big bang cosmology, which are properly addressed in cosmic inflation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 1 – In view of the above, looking for alternative gravity theories is a justified course of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1 The search for these so-called modified theories of gravity is in essence a search for healthy field equations that differ from those of Einstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Owing to Lovelock and his undisputed theorem [10, 11], there is a list of assumptions that we need to break (in one or more ways) in order to find such a set of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In particular, one of the assump- tions is that space-time is a smooth Lorentzian manifold equipped with a time orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Therefore, we can dodge the stringent consequences of the theorem by permitting the affine connection to be an independent field variable beyond the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A general connection has both torsion and non-metricity, and a gravitational theory for the metric and the affine connection is known as a Metric-Affine Gravity (MAG) theory [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' MAG theories exhibit many attractive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' First, the presence of a new gravitational potential, the independent affine connection, brings gravity conceptually closer to the other interactions whose mediators are gauge connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2 Second, an intriguing feature of metric-affine theories of gravity is the emergence of a hypermomentum current [13–16] in the presence of matter couplings to the gauge connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This differential form, obtained by varying the matter action with respect to the gauge connection, can be decomposed into the irreducible spin, dilation and shear parts which ought to excite the post-Riemannian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the above sense, MAG theories bring forth an astonishing interplay between matter with non-trivial microstructure and non- Riemannian effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, note that an interesting discussion has been revived about the status of MAG as a quantum theory though definitive conclusions are yet far from being drawn (see [17–19] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3 As with Riemannian theories, one is particularly interested in MAG theories which are solvable, ideally exactly solvable (at least for some symmetry ansatz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' If the task of finding exact solutions in Riemannian theories is in most cases a difficult one, then the trouble gets double in MAG because we also have to determine the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In fact, perhaps the most persistent obstruction to obtaining exact solutions with non-vanishing torsion and non-metricity in metric-affine theories,4 is the computational complexity one is bound to face when attempting to solve the field equations for the affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the dominant part of the MAG literature, the strategy to make the connection dynamical is, roughly put, to consider (at least) quadratic curvature invariants like R2, RµνRµν, RλρµνRλρµν, et cetera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This strategy is indeed well-motivated and fairly general, but it can quickly turn any attempt at finding a solution into a nearly impossible task, even for relatively simple (in form) Lagrangians of this sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The reason behind this is that the affine connection is a very compact package of a large number of degrees of freedom, the dynamics of which are encoded in the components of a single tensor equation and presented in an awfully coupled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In fact, among other techniques, one almost always tries to split this master tensor equation into simpler, hopefully decoupled equations by acting on 1For a review of the zoo of modified-gravity theories see [7–9] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 2See the notion of affine gauge theory in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 3See also [20–34] for some recent advances in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 4See [35–39] and references therein for some examples of black hole solutions with torsion and/or non- metricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 2 – it with some symmetry projector, or taking traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Therefore, it may not always be the case that the affine connection is the optimal field variable, beyond the metric, to describe a MAG theory, at least not for all intents and purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In this work, we embrace this point of view and use it as a motivation for our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Our goal is to make a change of field variables that will allow us to trade the connection field equations for an equivalent “decongested” system of simpler field equations obtained by letting an action vary with respect to tensor and vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' These tensor and vector fields, used to describe MAG theories anew, will be the irreducible pieces of torsion and non-metricity under the Lorentz group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Ipso facto, they are identified with the fundamental fields, the metric and the affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We then work out a complete mapping between the two frameworks which can be later used as a dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The advantage of the new framework is that we can now handle MAG theories as Riemannian theories with additional fields, at least within the context of the variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' These additional fields are part of the space-time geometry it self, and not some external entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' With the new framework established, we proceed with giving examples of how to con- struct MAG theories in vacuum which result in a selective and tractable self-excitation of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Although there is no universal prescription, a basic idea underlies all our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We take Riemannian theories with additional fields (vectors and tensors), which we exactly know how to solve, and we cast them, after some minor necessary modifications, into MAG theories which effectively yield the same field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The role of the addi- tional fields is now performed by the new field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Their propagation is tantamount to the excitation of (part of) the post-Riemannian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Even though the form of the metric solution in such MAG theories will, more or less, be already known in the gravity literature, the full solution, including the connection, will be novel, for it will in general feature non-zero torsion/non-metricity backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Plan of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In section 2 we convey the bare minimum in metric-affine theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Then, in section 3 we present the alternative framework and a detailed mapping between the latter and the ordinary Palatini approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Using the new framework, we revisit the Hilbert-Palatini action in section 4 hoping for fresh insight, and we introduce a useful variant of the new framework when projective symmetry is at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, in section 5 we showcase a series of examples where we apply the previously developed frameworks, and we also report solutions therein, concluding in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 2 Preliminaries This section is devoted to a brief communication of the MAG preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In metric-affine theories the affine connection is an independent field variable beyond the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We use it to define a covariant derivative whose action on vector and co-vector fields is given by ∇µV ν = ∂µV ν + Γν λµV λ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1a) ∇µVν = ∂µVν − Γλ νµVλ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1b) – 3 – where Γλ µν are the connection symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A general affine connection features both torsion and non-metricity given by T λµν = 2Γλ [νµ], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2a) Qλµν = −∇λgµν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2b) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The former introduces twisting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' parallel transport along a closed path results in a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The latter measures the failure of the metric to be covariantly constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' parallel transport brings about a change in vector norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Out of torsion and non-metricity we can construct three vectors and one axial tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Regarding torsion, we have the vector Tµ = T λµλ and the axial tensor Sα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='αn−3 = − 1 6(n − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='˜ϵα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='αn−3λµνTλµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3) Here, ˜ϵα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='αn = √−gϵα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='αn with ϵα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='αn being the Levi-Civita symbol in n space-time di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Our convention for the symbol is ϵ01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='n−1 = 1 = −ϵ01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In n = 4 dimensions, the above axial tensor is known as the torsion pseudo-vector, Sα = −1 6˜ϵαλµνTλµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) Regarding non-metricity, we have the vector Qµ = Qµαβgαβ, which is proportional to what is often called the Weyl vector in MAG lore, and ˇQµ = Qαβµgαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Continuing, we define the curvature tensor of the general affine connection as Rµναβ = ∂αΓµ νβ + Γµ ραΓρ νβ − α ↔ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5) From the above we can form three independent contractions, Rνβ = Rµνµβ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6a) ˆRαβ = Rµµαβ = ∂[αQβ], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6b) ˇRλα = Rλµανgµν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6c) which go by the name Ricci tensor, homothetic-curvature tensor, and co-Ricci tensor, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Notice that only the last contraction requires a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, contracting indices once more with the metric, we form the Ricci scalar R = Rµνgµν = ˇRµµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As per tradition, we will refer to the curvature tensor associated with the Levi-Civita connection as the Riemann tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Its single (double) trace will bear the name Riemannian Ricci tensor (scalar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Furthermore, it is a well-established fact that every affine connection differs from an- other affine connection by a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Therefore, we can always write a general affine connec- tion as Γλ µν = ˜Γλ µν + Nλµν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7) where ˜Γλ µν = 1 2gρλ (∂µgνρ + ∂νgµρ − ∂ρgµν) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8) – 4 – are the Christoffel symbols, and Nλµν = 1 2gρλ (Qµνρ + Qνρµ − Qρµν − Tρµν − Tνµρ − Tµνρ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) is the so-called distortion tensor encompassing the non-Riemannian DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Torsion and non- metricity can always be traded for the distortion tensor via the relations T λµν = −2Nλ[µν] and Qλµν = 2N(µν)λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7) suggests that we can split off any quantity into a Riemannian part and non-Riemannian contributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' this is the reputed post-Riemannian expansion of a quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' For instance, the post-Riemannian expansion of the curvature tensor reads Rµναβ = ˜Rµναβ + 2 ˜∇[αNµ |ν|β] + 2Nµ λ[αNλ |ν|β], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) where ˜∇α is the Levi-Civita covariant derivative and ˜Rµναβ the Riemann tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Unless otherwise stated, quantities with a tilde accent will always stand for objects associated with the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 3 The alternative framework Observe that the presence of an affine connection as an independent field variable introduces n3-many additional a priori DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Undeniably, the affine connection, being an essential constituent of the metric-affine geometry, is a meaningful variable to work with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' torsion and non-metricity are after all properties of a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' However, squashing that many degrees into a single field is not always the most convenient option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In this section, we instead distribute them among seven fields which correspond to the irreducible pieces of torsion and non-metricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This strange way of re-organizing the connection DoF will be suitable for purposes presented during a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The new fields will of course be identified with the metric and the affine connection, the fundamental field variables in metric-affine theories, thus allowing us — via this change of field variables — to describe any MAG theory anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We will show in full generality that the field equations derived within this new framework imply and are implied by the field equations obtained in the familiar context of the Fundamental (or Palatini) Framework (FF) where the metric and the affine connection are the independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The freedom to switch between different formulations of the same theory will prove to be a great asset in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In what follows, ˚aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' denotes the completely traceless part of a tensor aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', whereas ¯aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' denotes the complement of ˚aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' in aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', ¯aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' = aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' −˚aλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='. The irreducible decomposition of the torsion tensor under the Lorentz group yields Tλµν = Hλµν +˚tλµν + ¯tλµν, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1) where Hλµν = T[λµν], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2a) ˚tλµν = Tλµν − Hλµν − ¯tλµν, ¯tλµν = 2 n − 1gλ[νTµ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2b) – 5 – Note that instead of the 3-form field Hλµν one may alternatively use the dual tensor Sα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='αn−3 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Similarly, for non-metricity we have Qλµν = ˚πλµν + ¯πλµν + ˚qλµν + ¯qλµν, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3) where ˚πλµν = Q(λµν) − ¯πλµν, ¯πλµν = 1 n + 2g(λµρν), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4a) ˚qλµν = Qλµν − πλµν − ¯qλµν, ¯qλµν = 2 3(n − 1) � gλ(µuν) − gµνuλ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4b) uµ = ˇQµ − Qµ, ρµ = 2 ˇQµ + Qµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4c) Using the defining eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2), equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) tell us how to express the irre- ducible pieces in terms of the metric and the affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The other way around, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3) tell us how to express the affine connection in terms of the metric and the irreducible pieces using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since we will work with many fields, we find it befitting to use multi-field notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Let us introduce two objects, O and A, with components ON λµν and AI µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' They are given by Oλµν = {Hλµν, tλµν, πλµν, qλµν} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5a) Aµ = {Tµ, ρµ, uµ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5b) Einstein’s summation convention will also be adopted for indices M, N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', which take values in {1, 2, 3, 4}, and for indices I, J, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', which take values in the subset {2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5 We can lower/raise these indices with the reference metrics δMN and δIJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As above, whenever the capital indices are omitted, the objects should be understood as column vectors in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, the term Alternative Framework (AF) will be coined for the formulation of a MAG theory in terms of the set {g, ˚ ON, AI} of field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' With all the necessary ingredients at our disposal, let us consider a general n-dimensional MAG action in the FF, say I[g, Γ] = � √−gdnxL, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6) where g ≡ det g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We let it vary in order to get δI = � √−gdnx � Eµνδgµν + ∆λµνδΓλ µν � + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7) where s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' denotes the surface terms arising from integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We have also abbreviated the functional derivatives as Eµν = 1 √−g δI δgµν , ∆λµν = 1 √−g δI δΓλµν (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8) 5Note the use of slanted numerals for the value of an internal index as opposed to µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='.n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 6 – The field equations read Eµν = 0, ∆λµν = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) with Eµν being a symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6 On the other hand, considering eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3), we can write the previous action in the AF, namely I � g, ˚ ON, AI� = � √−gdnxL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) Letting I vary we get δI = � √−gdnx � ˆEµνδgµν + ˚ Oλµν N δ˚ ON λµν + Aµ I δAI µ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11) plus surface terms where ˚ ON and ˚ ON belong to the same irreducible tensor subspace as Lorentz tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The field equations read ˆEµν = 0, ˚ Oλµν N = 0, Aµ I = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='12) where ˆEµν is a symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Observe that the traceless property of ˚ ON λµν must be preserved when the action is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This condition can be enforced with a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The result is equivalent to simply demanding that the functional derivative with respect to ˚ ON λµν, ˚ Oλµν N , must be traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' With the above in hand, we turn our attention to finding the identities relating the functional derivatives { ˆE, ˚ ON, AI} to {E, ∆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' These identities will arise via identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Expressing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7) in terms of the AF variables, one finds that ˚ Oλµν 1 = −1 2∆[λµν], ˚ Oλµν 2 = ˚ D[µν]λ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13a) ˚ Oλµν 3 = 1 2 ˚∆(λµν), ˚ Oλµν 4 = −˚ Dλ(µν), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13b) Aµ 2 = 1 n − 1 � ∆µλλ − ∆λµλ� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13c) Aµ 3 = 1 6(n + 2) � ∆λλµ + ∆λµλ + ∆µλλ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13d) Aµ 4 = 1 3(n − 1) � 2∆µλλ − ∆λλµ − ∆λµλ� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13e) where ˚ D and ˚∆ are given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2) of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, we also have ˆEµν = Eµν − ∆α(µν) − δα (µ∆ν)ββ n − 1 � n − 1 6(n + 2)ρα + Tα + 2 3uα � + 2∆[αβ]β n − 1 ˚t(µν)α − −2∆(αβ)β + ∆ββα 2(n + 2) ˚πµνα + ∆αββ − ∆β(αβ) n − 1 ˚qαµν − −1 2∆(µ αβ � Hν)αβ − ˚πν)αβ + 2˚qν)αβ + 2˚t|βα|ν) � + +1 2 � ˜∇α∆(µν)α + ˜∇α∆(µ|α|ν) − ˜∇α∆α(µν) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) 6The delicate issue of surface-term handling is out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We rather assume that one has by all means ensured that the variational problem is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 7 – where we used the identities ˚t[µν]λ = −1 2 ˚tλµν, ˚q(µν)λ = −1 2˚qλµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='15) If we let the fields ˚ ON and AI on the right hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) denote expressions involving the metric and the connection symbols (see eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4)), the above simply gives us ˆE in terms of the FF quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' At this stage, we find it useful to display the “inverted form” of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) by expressing ∆ in terms of ˚ ON and AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Using the identity ˚ D[µν]λ = ˚ D[µ|λ|ν] − ˚ Dλ[µν], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16) we directly obtain ˚∆[λµν] = −2˚ Oλµν 1 , ˚∆(λµν) = 2˚ Oλµν 3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a) ˚ Dλ[µν] = −2 � ˚ O[µν]λ 4 + ˚ Oλµν 2 � , ˚ Dλ(µν) = −˚ Oλµν 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17b) The last three equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) form a separate matrix subsystem, invertible for n > 1, whose inversion yields ∆λλµ = (n − 1) � Aµ 2 − 2Aµ 4 � + 2(n + 2)Aµ 3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='18a) ∆λµλ = (n − 1) � Aµ 4 − Aµ 2 � + 2(n + 2)Aµ 3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='18b) ∆µλλ = (n − 1)Aµ 4 + 2(n + 2)Aµ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='18c) Recalling that the irreducible decomposition of a general rank-3 tensor has the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1), it all boils down to the equation ∆λµν/2 = ˚ Oλµν 3 − ˚ Oλµν 1 + ˚ Oνµλ 4 − ˚ Oλµν 2 + +3A(λ 3 gµν) + gν(λAµ) 4 − gµλAν 4 + gλ[µAν] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='19) Finally, one may take the above result, plug it into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14), and write the latter as Eµν = ˆEµν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='20) which provides us with E in terms of AF quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally note that the vanishing of a tensor implies that all its irreducible pieces vanish separately, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Let us now mold all these technical details into the main result we wish to convey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' When the field eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='12) hold true, we have that ∆λµν = 0 via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='19) and Eµν = 0 via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='20), ergo, the field eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) are implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The other way around, when the field eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) hold true, we have ˚ Oλµν N = 0 and Aµ I = 0 via eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14), it further follows that ˆEµν = 0, ergo, the field eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='12) are implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Consequently, we have shown an equivalence relation in detail, in particular, that the field equations in the two formulations imply and are implied by each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We also remark that, having the field equations in one of the two frameworks, it is always possible to reconstruct the field equations in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 8 – Lastly, having set up the new framework, we find it useful to report an interesting correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' There exist certain linear connection transformations in the FF which amount to translations of only one irreducible piece at a time (preserving the rest) in the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Before disclosing them, let us bring yet another pair of multi-fields to our aid, o and a, with components oN λµν and aI µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Note that ˚oN and ˚ ON belong to the same irreducible tensor subspace as Lorentz tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' After some straightforward algebra we arrive at a 1:1 correspondence between the translations ˚ O′N = ˚ ON +˚oN, A′I = AI + aI, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21) in the AF (space-time indices understood, thus omitted) and the linear connection trans- formations Γ′λ µν = Γλ µν + (δΓ)λ µν with (δΓ)λµν = −1 2˚o1 λµν, (δΓ)λµν = −˚o2 νµλ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22a) (δΓ)λµν = 1 2˚o3 λµν, (δΓ)λµν = −˚o4 λµν, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22b) (δΓ)λµν = 2 n − 1gν[µa2 λ], (δΓ)λµν = 1 2(n + 2)a3 (λgµν), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22c) (δΓ)λµν = 2 3(n − 1) � gµνa4 λ − gλ(µa4 ν) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22d) in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We also report that under a local Weyl re-scaling of the metric, g′ µν = e−2φ(x)gµν, the tensor fields ˚ ON λµν must have conformal weight −2, and thus, transform as the metric, whereas T ′ µ = Tµ, ρ′ µ = ρµ + 2(n + 2)∂µφ, u′ µ = uµ − 2(n − 1)∂µφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='23) Clearly, the combination ρµ +(n+2)uµ/(n−1) is itself a Weyl invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' It corresponds to 3(n ˇQµ − Qµ)/(n − 1) in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We shall now proceed with a highly pedagogical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 4 Revisiting the Hilbert-Palatini action 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1 FF vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' AF The n-dimensional Hilbert-Palatini (HP) action reads IHP = 1 2 � √−gdnxR, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1) in units ℏ = c = MPl = 1 where MPl is the reduced Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This is the standard FF action which is invariant under the so-called projective transformation Γ′λ µν = Γλ µν + 1 n − 1δλ µξν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2) with ξµ being an arbitrary vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 9 – The field equations in the FF read 2Eµν ≡ R(µν) − 1 2Rgµν = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3a) 2∆λµν ≡ δν λNµαα − Nµνλ − Nνλµ + Nαλαgµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3b) We chose to express the connection field equations in terms of the distortion tensor in order to achieve a more compact output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The invariance of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3) under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2) can be easily seen from the fact that R′ µν = Rµν + 2 n − 1∂[µξν], ∆λλµ ≡ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) the right one holding true identically (off-shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' It is a well-known fact that the solution to ∆λµν = 0 is the affine connection Γλ µν = ˜Γλ µν + δλ µVν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5) where Vµ is some undetermined vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since Γλ µν = ˜Γλ µν + δλ µ � Vν + 1 n − 1ξν � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6) is also a solution, we conclude that the affine connection solving the connection field equa- tions is just the Levi-Civita connection up to the choice of gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The effective form of the metric field equations becomes ˜Rµν = 1 2 ˜Rgµν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', the HP action is effectively Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' On the other hand, in the AF, whenever we write R we just mean the expression ˜R + RT + RV + ˜∇µ (2T µ + uµ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8) where ˜R is the Riemannian Ricci scalar and RT = −1 4H2 − 1 4˚π2 + 1 2˚qλµν˚qλµν + 1 2 ˚tλµν˚tλµν + ˚qλνµ˚tνµλ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9a) RV = n − 1 36(n + 2)ρ2 − n − 2 n − 1T 2 + 5 − 2n 9(n − 1)u2 + 1 18ρµuµ − n − 2 n − 1Tµuµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9b) Therefore, up to surface terms, our AF action reads IHP = 1 2 � √−gdnx � ˜R + RT + RV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) The analogue of a projective transformation in the AF is comprised of the simultaneous translations A′I = AI + aI−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11) with ξµ ≡ a1 µ = n − 1 2(n + 2)a2 µ = −1 2a3 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='12) – 10 – One can easily verify that the above transformations should only affect RV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since it hap- pens that RV is invariant, the transformations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11) constitute a symmetry of the full action (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Now, there are two equivalent ways to proceed as we have shown in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We can either use eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3) to reconstruct the field equations in the AF, or we can directly vary the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) with respect to the AF field variables (quickest strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Both methods lead to the same result, namely the field equations ˚ Oλµν 1 ≡ −1 4Hλµν = 0, ˚ Oλµν 2 ≡ 1 2 � ˚tλµν − ˚q[µν]λ� = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13a) ˚ Oλµν 3 ≡ −1 4˚πλµν = 0, ˚ Oλµν 4 ≡ ˚ O(µν)λ 2 + 1 8˚qλµν = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13b) Aµ 2 ≡ −n − 2 n − 1 � T µ + 1 2uµ � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13c) Aµ 3 ≡ n − 1 36(n + 2) � ρµ + n + 2 n − 1uµ � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13d) Aµ 4 ≡ 1 2Aµ 2 + n + 2 n − 1Aµ 3 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13e) and 2 ˆEµν ≡ ˜Rµν − 1 2gµν � ˜R + RT + RV � − 3 4 � HµαβHναβ + ˚πµαβ˚πναβ� + +˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 2 � ˚qµαβ˚qναβ +˚tµαβ˚tναβ� + +˚tαβ(µ˚qν) αβ − ˚qαβ (µ˚tν)αβ −˚tβα(µ˚qαβ ν) − −n − 2 n − 1TµTν + n − 1 36(n + 2)ρµρν + 5 − 2n 9(n − 1)uµuν − −n − 2 n − 1T(µuν) + 1 18ρ(µuν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) It is evident that the first two lines in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) suggest that ˚ ON λµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The remaining two independent equations, Aµ 2 = 0 = Aµ 3, do further imply that uµ = −2Tµ = −n − 1 n + 2ρµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='15) Hence, the full solution to the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) is ˚ ON λµν = 0, Vµ ≡ uµ = −2Tµ = −n − 1 n + 2ρµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16) where Vµ is an arbitrary vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since Tµ = −1 2Vµ + ξµ, ρµ = n + 2 n − 1 (2ξµ − Vµ) , uµ = Vµ − 2ξµ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17) is also a solution, we conclude that AI µ = 0 up to the choice of gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The effective form of the metric field equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) becomes ˜Rµν = 1 2 ˜Rgµν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='18) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', the HP action in the AF is again, in effect, Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 11 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2 Projective symmetry and the AF◦ The careful reader would have already noticed that what is a projective symmetry in the FF manifests itself as a true gauge symmetry in the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Indeed, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) reveal that there are only two independent equations for the triplet Aµ which rather signals that one of these field variables is after all redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' First, bear in mind that the combinations Tµ + 1 2uµ, ρµ + n + 2 n − 1uµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='19) which are invariant under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11), correspond to the FF combinations 1 2 � ˇQµ − Qµ � + Tµ, 3 n − 1 � n ˇQµ − Qµ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='20) respectively, which are invariant under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Let us then discuss the idea that for n > 2, whenever the projective symmetry is at play, one should favor a doublet Bµ = �2 3 � Tµ + 1 2uµ � , n − 1 9 √ n2 − 4 � ρµ + n + 2 n − 1uµ �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21) over the redundant triplet Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Note that the above choice of BA, where indices A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' assume values in {1, 2}, is not the most general one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Nevertheless, it is the most convenient choice for our purposes here since it casts RV into the neat form RV = 9(n − 2) 4(n − 1)ηABBA µ BB ν gµν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22) where ηAB are the components of the two-dimensional Minkowski metric η(2) = diag(−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The affected parts of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) read Aµ 2 ≡ −3(n − 2) 2(n − 1)Bµ 1 = 0, Aµ 3 ≡ 1 4 � n − 2 n + 2Bµ 2 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='23) whereas the metric field equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) are rendered into 2 ˆEµν ≡ ˜Rµν − 1 2gµν � ˜R + RT + RV � − 3 4 � HµαβHναβ + ˚πµαβ˚πναβ� + +˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 2 � ˚qµαβ˚qναβ +˚tµαβ˚tναβ� + +˚tαβ(µ˚qν) αβ − ˚qαβ (µ˚tν)αβ −˚tβα(µ˚qαβ ν) + 9(n − 2) 4(n − 1)ηABBA µ BB ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='24) Interestingly, when written in terms of the BA fields, the HP action and its field equations exhibit an SO(1, 1) symmetry!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Indeed, the field transformation B′ µ = Λ(x)Bµ with Λ = � cosh θ(x) sinh θ(x) sinh θ(x) cosh θ(x) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='25) preserves both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We remark that the manifestation of this transformation as a group action on the field variables is exclusive to the use of the BA’s to formulate the HP action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 12 – Since the use of these field re-combinations revealed something new, we find it worth to take a step back and generalize the whole thing to another framework which we dub AF◦ or “diminished alternative framework”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the AF◦, we formulate our MAG theory in terms of the reduced set {g, ˚ ON, BA} of field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The doublet Bµ should consist of two linear combinations of AF vector fields which are invariant under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Equivalently, it should be comprised of two linear combinations of Tµ, Qµ, ˇQµ which are invariant under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2), the point being that in the AF◦ such transformations should constitute an identity operation on our field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The most general combinations invariant under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11) are B1 µ = αI−1AI µ, B2 µ = βI−1AI µ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='26) with α3 = α1 2 + (n + 2)α2 n − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='27) and ditto for the coefficients βA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Let Bµ A = 1 √−g δI δBA µ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='28) such that the field equations for the field BA are Bµ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We can directly make the identifications Aµ I = αI−1Bµ 1 + βI−1Bµ 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='29) where one has to remember that the coefficients obey the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Clearly, whenever Bµ A = 0, it follows that Aµ I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' However, whenever Aµ I = 0, it follows that Bµ A = 0 only when α1β2 − α2β1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='30) Therefore, the field equations in the AF◦ imply and are, under assumptions, implied by the field equations in the AF or in the FF (if we follow the equivalence chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Lastly, let us see exactly how we ended up with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In terms of the fields BA, as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='26), we have that RV = � f(β2 1, β2 2)B1 µB1 ν − 2f(α1β1, α2β2)B1 µB2 ν + f(α2 1, α2 2)B2 µB2 ν � gµν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='31) where f(x, y) := (n − 1)2x − 36(n2 − 4)y 36(n2 + n − 2)(α2β1 − α1β2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='32) Moreover, the total divergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8) assumes the form 2 α2β1 − α1β2 ˜∇µ (α2Bµ 2 − β2Bµ 1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='33) To get the above, we expressed α3 in terms of α1, α2 via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='27), and ditto for the parameters βA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Different choices for the parameters αA, βA obviously amount to different changes of field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 13 – A convenient choice is one for which f(α1β1, α2β2) = 0, namely β2 = (n − 1)2α1β1 36(n2 − 4)α2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='34) provided n > 2, which yields RV = (n − 2)(n − 1) (n − 1)2α2 1 − 36(n2 − 4)α2 2 � −B1 µB1 ν + 36α2 2(n2 − 4) β2 1(n − 1)2 B2 µB2 ν � gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='35) Further imposing that β1 = 6|α2| √ n2 − 4 n − 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='36) gives RV = (n − 2)(n − 1) (n − 1)2α2 1 − 36(n2 − 4)α2 2 ηABBA µ BB ν gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37) Finally, we choose α2 = n − 1 18 � 9α2 1 − 4 n2 − 4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='38) for later convenience, which leads to RV = 9(n − 2) 4(n − 1)ηABBA µ BB ν gµν =: RV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='39) Note that all of the above parameter choices are in agreement with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='30) which becomes 2(n − 1) 27 √ n2 − 4 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='40) In terms of the AF fields, our new field variables, BA, read B1 µ = α1Tµ + n − 1 18 � 9α2 1 − 4 n2 − 4 ρµ + 1 18 � �9α1 + � (9α2 1 − 4)(n + 2) n − 2 � � uµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='41a) B2 µ = � 9α2 1 − 4 3 Tµ + (n − 1)α1 6 √ n2 − 4 ρµ + 1 6 �� n + 2 n − 2α1 + � 9α2 1 − 4 � uµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='41b) where we may further fix |α1| = 2/3 so that B2 is purely a combination of traces of the non-metricity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This brings us to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Henceforth, the word AF◦ will always mean that we use the specific doublet (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3 SO(1, 2) symmetry, torsion/non-metricity rotations, and the d-AF◦ Now, we restrict ourselves to n = 4 space-time dimensions where things get a bit more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Via the dualization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4), we have a pseudo-vector, and the term ∝ H2 can be moved from RT to RV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In particular, let us introduce the objects ˆRT = −1 4˚π2 + 1 2˚qλµν˚qλµν + 1 2 ˚tλµν˚tλµν + ˚qλνµ˚tνµλ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='42a) ˆRV = 3 2ηABBA µ BB ν gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='42b) – 14 – where the calligraphic indices take values in {1, 2, 3}, ηAB are the components of the three-dimensional Minkowski metric, η(3) = diag(−1, 1, 1), and we have formed a triplet Bµ = � B1 µ, B2 µ, Sµ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='43) with BA given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The affected parts of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) read Aµ 2 ≡ −3(n − 2) 2(n − 1)Bµ 1 = 0, Aµ 3 ≡ 1 4 � n − 2 n + 2Bµ 2 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='44) and ˚ Oλµν 1 ≡ 1 4˜ϵλµναSα = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='45) whereas the metric field equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) are rendered into 2 ˆEµν ≡ ˜Rµν − 1 2gµν � ˜R + ˆRT + ˆRV � − 3 4˚πµαβ˚πναβ + +˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 2 � ˚qµαβ˚qναβ +˚tµαβ˚tναβ� + +˚tαβ(µ˚qν) αβ − ˚qαβ (µ˚tν)αβ −˚tβα(µ˚qαβ ν) + 3 2ηABBA µ BB ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='46) Remarkably, when written in terms of the triplet (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='43), the four-dimensional HP action and its field equations exhibit a larger symmetry under an SO(1, 2) group action mixing the components BA µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Of particular interest is the transformation B′ µ = Λ(x)Bµ with Λ = � � � 1 cos θ(x) sin θ(x) − sin θ(x) cos θ(x) � � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='47) which represents an SO(2) rotation in the {B2 µ, Sµ} (field) subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Specifically, the invariance of the action under the discrete transformation with Λ(θ = π/2), constitutes an exceptional example of a symmetry under torsion/non-metricity rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Indeed, omitting the space-time indices, we have � � � B1 B2 S � � � → � � � B1 S −B2 � � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='48) where Sµ is pure torsion (pseudo-vector) and B2 µ is pure non-metricity (traces), as de- fined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Do further note that if we consider B2 µ and Sµ as, respectively, the real and imaginary parts of a complex vector τµ = B2 µ + iSµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='49) we have that ˆRV = −3 2 � B1 µB1 ν − τµτ ∗ ν � gµν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='50) – 15 – where τ ∗ is the complex conjugate of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The previous SO(2) symmetry now manifests itself as a U(1) under τ ′ µ = e−iθ(x)τµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As a closing remark, let us mention here that in what follows we will often use the tor- sion pseudo-vector Sµ instead of Hλµν as a field variable in our four-dimensional projective- invariant examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In other words, we will be using the set {g, ˚ OI, BA} when invoking the AF◦, and some clarifications are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Letting an action I = � √−gd4xL[g, ˚ OI, BA] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='51) vary, we get δI = � √−gd4x � ˇEµνδgµν + ˚ Oλµν I δ˚ OI λµν + Bµ AδBA µ � + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='. (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='52) Therefore, the field equations read ˇEµν = 0, ˚ Oλµν I = 0, Bµ A = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='53) where ˇEµν is a symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Following the preceded steps, one should be able to directly make the identifications Bµ 1 = 3 2Aµ 2, Bµ 2 = 6 √ 3Aµ 3, Bµ 3 = ˚ O1 αβγ˜ϵµαβγ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='54a) Aµ 4 = 1 3 � Bµ 1 + 1 √ 3Bµ 2 � , ˇEµν = ˆEµν + B3 (µSν) − 1 2gµνSαBα 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='54b) which tell us how the functional derivatives in the two frameworks are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' These do once again suffice to prove the equivalence between the field equations in the AF and in this framework which, for the sake of clarity and in lack of a better name, we call d-AF◦, with the d reminding us that we use the dual of Hλµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 5 Exciting the connection: a series of examples It has been standard practice in the MAG community to motivate actions from a geometric perspective and with full generality in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Although this is in general good practice, it often leads to an intractable set of field equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' in the end one will unavoidably sacrifice generality to get results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In this section we propose an different motivation for MAG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Using the alternative frameworks we presented, we write meaningful field theories propagating some of the new field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' These are, in essence, MAG theories propagating certain connection DoF in a tractable and controllable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' These MAG theories are inspired by Riemannian theories with additional fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In fact, they yield an effective7 set of field equations which is very much identical to the corresponding system of equations in the Riemannian case, the crucial difference being that instead of additional fields we use specific modes of the distortion tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The obvious 7The word “effective” is used to denote that one has eliminated auxiliary field variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', fields which do not appear with time derivatives in the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 16 – advantage is that for a given symmetry ansatz, we exactly know how to solve the differential equations that arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Hence, one should not expect metric solutions novel in form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' the only novelty is that these known metric backgrounds are now part of a larger solution with torsion and non-metricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In what follows, we will occasionally omit space-time indices (or internal indices) when they are trivially understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1 The MAGswell theory The action for a Maxwell field Aµ coupled to four-dimensional gravity with a cosmological constant is ˜IEM = 1 2 � √−gd4x � ˜R − 2Λ − 1 2F 2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1) where Fµν = 2∂[µAν] is the field-strength tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The above integral is invariant under shifts A′ = A + dθ, where θ(x) is some scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Variation with respect to the metric yields the metric field equations ˜Gµν + Λgµν = FµαFνα − 1 4F 2gµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2) where ˜Gµν = ˜Rµν − 1 2 ˜Rgµν is the Einstein tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The Maxwell equations and the Bianchi identity can be written as ∂ν �√−gF νµ� = 0, ∂ν �√−g ∗ F νµ� = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3) respectively, where ∗F µν = 1 2˜ϵµνρλFρλ is the Hodge dual of the field-strength tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Here, we wish to write down a more or less similar theory for a massless vector in MAG, the homophonous MAGswell field, Cµ, as we may playfully dub it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We will do so by considering the Ricci scalar as our cornerstone and adding proper terms to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since the HP action is invariant under projective transformations we would like to retain this feature in the complete action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This will also allow us to work in the AF◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We emphasize that the MAGswell field should be understood as part of the geometry of space-time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', it is not the familiar gauge connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Without further ado, let us consider a fairly general projective-invariant candidate for the four-dimensional MAGswell action which in the d-AF◦ assumes the form IC = � √−gd4xLC ≡ � √−gd4x (LHP + Lct + Lkin) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) with LHP = 1 2 (R − 2Λ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5a) Lct = −3 4gµνBA µ BB ν ηAB, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5b) Lkin = −1 4F 2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='5c) Here, F (C) µν = 2∂[µCν] with Cµ ≡ αABA µ being the composite MAGswell field and αA dimensionless constants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', real non-zero numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The curvature scalar R stands for ˜R + ˆRT + ˆRV + 3 ˜∇µBµ 1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='6) – 17 – with the constituents given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The form of the action in the AF, or the FF, can be easily obtained by remembering that B1 µ = 1 3 (2Tµ + uµ) = 1 3 � ˇQµ − Qµ + 2Tµ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7a) B2 µ = 1 6 √ 3 (ρµ + 2uµ) = 1 6 √ 3 � 4 ˇQµ − Qµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7b) For example, in the FF we have Lct = − 1 36 ˇQµ ˇQµ − 1 9 ˇQµQµ + 11 144QµQµ + 1 3 � ˇQµTµ − QµTµ + TµT µ� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='8) and Lkin is a linear combination of F 2 (T), F 2 (Q), F 2 ( ˇQ), F (T) µν F µν (Q), F (T) µν F µν ( ˇQ) and F (Q) µν F µν ( ˇQ), where F (T) µν = 2∂[µTν], F (Q) µν = 2∂[µQν], F ( ˇQ) µν = 2∂[µ ˇQν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) We remind the reader that we work in natural units with the reduced Planck mass further set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A quick inspection of the action indicates that the role of Lct is that of a counter-term Lagrangian introduced to eradicate the mass terms for the BA µ ’s, entering via the Ricci scalar, in order for Cµ to appear massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Symmetries in the AF◦ (or d-AF◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) is invariant under the simulta- neous shifts B′A = BA + bA (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) with b2 µ = ∂µθ − α1b1 µ α2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11) where θ(x) is some real scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The number of free gauge parameters to be fixed is thus two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Symmetries in the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) in the AF is invariant under the simultaneous shifts A′I = AI + bI−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='12) with b3 µ = −12α1b1 µ + α2 √ 3b2 µ − 18∂µθ 6α1 + 2 √ 3α2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13) The number of free gauge parameters to be fixed is three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Symmetries in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) in the FF is invariant under the connection transformation Γ′λ µν = Γλ µν + aλgµν + δλ ν bµ + δλ µcν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14) with bµ = (α1 + α2 √ 3)aµ − ∂µθ α1 − α2 √ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='15) The number of free gauge parameters to be fixed is three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 18 – Indeed, if #fgp(−) outputs the number of free gauge parameters in a certain framework, then #fgp(FF) = #fgp(AF) in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The fact that #fgp(AF◦) = #fgp(AF) − 1 has to do with the projective-symmetry “charge” being initially absorbed into the field variables of the AF◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Having discussed the symmetries in the different frameworks, we now turn our attention to the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the d-AF◦, they read ˚ Oλµν 2 ≡ 1 2 � ˚tλµν − ˚q[µν]λ� = 0, ˚ Oλµν 3 ≡ −1 4˚πλµν = 0, ˚ Oλµν 4 ≡ ˚ O(µν)λ 2 + 1 8˚qλµν = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16a) Bµ 1 ≡ α1 α2 Bµ 2 ≡ α1 ˜∇νF νµ (C) = 0, Bµ 3 ≡ 3 2Sµ = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16b) 2 ˇEµν ≡ ˜Gµν − 1 2gµν � ˆRT − 2Λ + 3 2S2 − 1 2F 2 (C) � − 3 4˚πµαβ˚πναβ + +˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 2 � ˚qµαβ˚qναβ +˚tµαβ˚tναβ� + +˚tαβ(µ˚qν) αβ − ˚qαβ (µ˚tν)αβ −˚tβα(µ˚qαβ ν) + 3 2SµSν − F (C) µ αF (C) να .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16c) The first three of them imply ˚ OI λµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Then, equation Bµ 3 = 0 suggests that Sµ = 0 which yields Hλµν = 0 via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Substituting these results back into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16), we obtain the effective set ∂ν �√−gF νµ (C) � = 0 = ∂ν �√−g ∗ F νµ (C) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a) ˜Gµν + Λgµν = F (C) µ αF (C) να − 1 4F 2 (C)gµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17b) where we took the liberty to also include the Bianchi identity for F (C) µν with ∗ F µν (C) = 1 2˜ϵµνρλF (C) ρλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='18) As differential equations, these exactly correspond to the Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Let us now study the solution in the different frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Solution in the d-AF◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As already mentioned, we have ˚ OI λµν = 0 = Sµ and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a), which only determines the combination Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' If ⟨Cµ⟩ + ∂µφ is the value of Cµ ≡ αABA µ satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a), then BA acquires the value ⟨BA⟩ with ⟨B2 µ⟩ = ⟨Cµ⟩ + ∂µφ − α1⟨B1 µ⟩ α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='19) Clearly, the values ⟨BA⟩ + bA, where the bA’s obey eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11), are also acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A good strategy to capture the solution in all possible gauges is to set b1 = −⟨B1⟩ + ˜α⟨C⟩, θ = −φ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='20) – 19 – where ˜α is a real number, obtaining B1 = ˜α⟨C⟩, B2 = 1 − ˜αα1 α2 ⟨C⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21) Therefore, {B1, B2} = � � � � � � � {0, ⟨C⟩/α2} ˜α = 0 {⟨C⟩/α1, 0} ˜α = 1/α1 {˜α⟨C⟩, (1 − ˜αα1)⟨C⟩/α2} ˜α ̸= 0, 1/α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22) Solution in the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The next step is to translate the solution into the language of the AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The field equations tell us that ˚ ON λµν = 0, and that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a) must hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Again, if ⟨Cµ⟩ + ∂µφ is the value of Cµ ≡ α2 6 √ 3ρµ + 2α1 3 Tµ + 3α1 + √ 3α2 9 uµ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='23) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a), then AI acquires the value ⟨AI⟩ with ⟨uµ⟩ = 18 (⟨Cµ⟩ + ∂µφ) − √ 3α2⟨ρµ⟩ − 12α1⟨Tµ⟩ 2(3α1 + α2 √ 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='24) Clearly, the values ⟨AI⟩ + bI−1, where the bI−1’s obey eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='13), are as good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Setting b1 = α⟨C⟩ − ⟨T⟩, b2 = β⟨C⟩ − ⟨ρ⟩, θ = −φ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='25) where α, β are real numbers, we get T = α⟨C⟩, ρ = β⟨C⟩, u = 18 − 12αα1 − βα2 √ 3 2(3α1 + α2 √ 3) ⟨C⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='26) We can further identify ˜α = 18 + √ 3(4α − β)α2 6(3α1 + α2 √ 3) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='27) and we collect the various cases in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Solution in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The final step is to translate the solution into the language of the familiar Palatini formalism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', to present an affine connection which solves the connection field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This reads Γλ µν = ˜Γλ µν + ⟨V λ⟩gµν − δλ ν ⟨Cµ⟩ + ∂µφ − � α1 + α2 √ 3 � ⟨Vµ⟩ α1 − α2 √ 3 + δλ µ⟨Uν⟩, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='28) where ⟨Cµ⟩ + ∂µφ is the value of Cµ ≡ 3α1 + 2 √ 3α2 9 ˇQµ − 6α1 + √ 3α2 18 Qµ + 2α1 3 Tµ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='29) – 20 – α β ˜α Tµ ρµ uµ 0 0 3 3α1 +α2 √ 3 0 0 ✓ 0 6 √ 3 α2 0 0 ✓ 0 3 2α1 0 1 α1 ✓ 0 0 ̸= 0, 3 2α1 2 √ 3(3−2αα1 ) α2 2α 3 ✓ ✓ 0 ̸= 0, 3 2α1 0 9+2 √ 3αα2 9α1 +3α2 √ 3 ✓ 0 ✓ 0 ̸= 0, 6 √ 3 α2 18− √ 3βα2 6(3α1 +α2 √ 3) 0 ✓ ✓ ̸= 0 ̸= 0, 2 √ 3(3−2αα1 ) α2 18+ √ 3(4α−β)α2 6(3α1 +α2 √ 3) ✓ ✓ ✓ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Values of ˜α and the AF field variables AI for different numbers α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A checkmark ✓ indicates that the ticked field is proportional to ⟨Cµ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='17a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Again, due to the freedom to shift our connection as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='14), and setting a = 24 − (4α − β)(α1 − α2 √ 3) 12(3α1 + α2 √ 3) ⟨C⟩ − ⟨V ⟩, θ = −φ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='30a) c = (8α + β)α2 1 − 4α1(3 + 2αα2 √ 3) − 3α2(βα2 − 4 √ 3) 12(3α2 1 − 2α1α2 √ 3 − 3α2 2) ⟨C⟩ − ⟨U⟩, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='30b) we reach a connection with torsion and non-metricity T λµν = 2α 3 δλ [ν⟨Cµ]⟩, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='31a) Qλµν = β 6 g(λµ⟨Cν)⟩ + 18 − 12αα1 − βα2 √ 3 9(3α1 + α2 √ 3) � gλ(µ⟨Cν)⟩ − gµν⟨Cλ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='31b) One can immediately verify that if we decompose the latter under the Lorentz group, we will exactly find (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='26) as the only excited irreducible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Therefore, one can again refer to table 1 for the various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' An interesting remark is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The Lagrangian does undeniably propagate the massless combination Cµ, a spin-1 geometric “boson”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This means that part of the post- Riemannian structure gets (self-)excited but it turns out to be impossible to make a gauge- independent statement about specifically which part that is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' For example, what appears to be an excitation of only torsional DoF in one gauge, shows up as an excitation of only non-metricity DoF in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Hence, propagation of the MAGswell field is tantamount to a self-excitation of the connection background with different parts of the latter being excited in the different gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Do also note that the action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) can be thought of as the massless limit of a massive theory which has an action like (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4), but with Lct replaced by Lmass = −1 2 � (µ2α2 1 − 3)B1 µB1 ν + 2µ2α1α2B1 µB2 ν + (µ2α2 2 + 3)B2 µB2 ν � gµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='32) – 21 – such that, up to surface terms, IC = 1 2 � √−gd4x � ˜R − 2Λ + ˆRT + 3S2 − µ2C2 − 1 2F 2 (C) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='33) always in the d-AF◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Obviously, Lct = Lmass(µ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The last two terms in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='33) imply that the combination Cµ behaves as a Proca field with mass µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since the HP action already introduces a mass scale proportional to the Planck mass,8 naturalness criteria suggest that we take µ to be of the same order (the composite field Cµ is part of the space-time geometry, not some external field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The field equations in the d-AF◦ are (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='16) except that ˜∇νF νµ (C) = 0 is replaced by the Proca equation ˜∇νF νµ (C) − µ2Cµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='34) Observe that the massive action and the field equations following from it, do still possess a symmetry under (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) if θA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This means that the propagated combination Cµ is a massive vector-boson, and the geometric interpretation of this propagation again falls into the previous scheme, viz, it is subject to the choice of gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, we remark that the MAGswell field is of course by itself not a solid and unique concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Nevertheless, let us justify why we think that Cµ is indeed the most general candidate to describe it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' First of all, playing the devil’s advocate, one could argue that there are more general projective-invariant combinations to take as our BA fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' we have already shown this in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Indeed, there is simply no physical argument favoring (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21) over (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Sure, the diagonal form of the mass-squared matrix and the emergence of SO symmetries under transformations in field space are nice features, but they are far from being necessary restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Actually, these features are completely absent here because (i) we have removed the mass terms, and (ii) there are no SO transformations being a symmetry of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' However, the real question is if we would gain more insight by considering a more complicated change of field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The answer is no, for Cµ would again be a linear combination of the AI’s but with different coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Assuming that we would also properly modify Lct as to remove algebraic instances of the new BA’s, it is evident that we would not get qualitatively different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Second, one could argue that projective invariance of the action is definitely not manda- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' One could instead add a kinetic term for any of the vector variables in the AF and remove all algebraic instances of this field from the action by introducing a proper counter- term Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This theory, which would no longer be invariant under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11) (the AF analogue of what is a projective transformation in the FF), would then propagate the gravi- ton and the specific mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Fortunately, one can easily prove that the solutions in all these different cases would correspond to the one and only solution in the theory with action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) in different gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2 Non-linear interacting MAG theory, black holes and solitons To showcase the usefulness of this new approach to MAG for obtaining exact solutions with torsion and non-metricity, let us propose a four-dimensional interacting action with 8Remember that we have set the reduced Planck mass to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 22 – non-linear dynamics for the MAGswell field Cµ and the pseudo-vector Sµ, namely INL = � √−gd4x �1 2(R − 2Λ − ˆRV ) − γ1F 2 (C) − γ2F 2 (S) − γLint � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='35) where Lint = δµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='µ4 ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='ν4 F (C) µ1µ2F (S) µ3µ4F ν1ν2 (C) F ν3ν4 (S) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='36) γ is a positive coupling constant of mass-dimension −4, and γ1, γ2 are positive coupling constants of mass-dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This is the form of the action in the d-AF◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The field strengths can be customarily written using only the partial derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', F(C) as previously defined and F (S) µν = ∂[µSν] with Sµ given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The term R stands for ˜R + ˆRT + ˆRV + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', with the constituents defined in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' One can always express the action in the FF (or the AF) by recalling eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This interacting Lagrangian was proposed in [40] for two distinct potentials in a Riemannian setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Here, we endow these potentials with a special geometric origin and cast the whole thing as a MAG theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the d-AF◦, the field equations read ˚ Oλµν 2 ≡ 1 2 � ˚tλµν − ˚q[µν]λ� = 0, ˚ Oλµν 3 ≡ −1 4˚πλµν = 0, ˚ Oλµν 4 ≡ ˚ O(µν)λ 2 + 1 8˚qλµν = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37a) Bµ 1 4α1 ≡ Bµ 2 4α2 ≡ γ1 ˜∇νF νµ (C) − γδµνρσ αβγδF (S) ρσ ˜∇ν � F αβ (C)F γδ (S) � = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37b) Bµ 3 4 ≡ γ2 ˜∇νF νµ (S) + γδνρσµ αβγδF (C) νρ ˜∇σ � F αβ (C)F γδ (S) � = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37c) 2 ˇEµν ≡ ˜Gµν − 1 2gµν � ˆRT − 2Λ − 2γ1F 2 (C) − 2γ2F 2 (S) + 2γLint � + +˚qαβµ˚qαβν +˚tαβµ˚tαβν + 1 2 � ˚qµαβ˚qναβ +˚tµαβ˚tναβ� + +˚tαβ(µ˚qν) αβ − ˚qαβ (µ˚tν)αβ −˚tβα(µ˚qαβ ν) − 3 4˚πµαβ˚πναβ − −4γ1F (C) µ αF (C) να − 4γ2F (S) µ αF (S) να = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37d) To get the above expressions, we also used the Bianchi identities dF(C) = 0 = dF(S) and the dimension-dependent identity δµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='µ4 ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='ν4 F (C) [µ1µ2F (S) µ3µ4F ν1ν2 (C) F ν3ν4 (S) gµ]ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='38) The action and the field equations are invariant under the transformation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='10) with the bA’s constrained via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' They are also invariant under a shift of Sµ by a locally exact co-vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', S′ µ = Sµ + ∂µφ, which in the AF amounts to H′ λµν = Hλµν − ˜ϵλµνα∂αφ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='39) whereas it corresponds to the connection transformation Γ′λ µν = Γλ µν + 1 2˜ϵλµνα∂αφ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='40) – 23 – in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Following the steps laid down in the previous section, we expect a connection solution with torsion and non-metricity T λµν = 2α 3 δλ [ν⟨Cµ]⟩ + ⟨Sα⟩˜ϵαλµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='41a) Qλµν = β 6 g(λµ⟨Cν)⟩ + 18 − 12αα1 − βα2 √ 3 9(3α1 + α2 √ 3) � gλ(µ⟨Cν)⟩ − gµν⟨Cλ⟩ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='41b) respectively, where ⟨C⟩ and ⟨S⟩ satisfy the non-linear differential equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37b) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We remind the reader that our gauge freedom is fully exhausted once we fix values for α, β (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Now, let us consider the static spherically-symmetric metric ansatz ds2 = −f(r)dt2 + dr2 f(r) + r2dΣ2 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='42) where dΣ2 2 = dχ2 + sin2 χdy2 gives the line element of a two-dimensional spherical section with χ, y compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We also make the following ansätze, Cµ = c(r)δ0 µ, Sµ = p cos χδ3 µ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='43) which result in F (C) µν = c′δ10 µν, F (S) µν = p sin χδ32 µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='44) A prime denotes differentiation with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Given the above, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37c) is identically satisfied, while eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37b) gives r(8γp2 + γ1r4)c′′ + 2(γ1r4 − 8γp2)c′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='45) This yields the first integral c′ = − qr2 γ1r4 + 8γp2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='46) where q is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Integrating once more, we get c = c0 + q γ1r 2F1 �1 4, 1, 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' −8γp2 γ1r4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='47) where 2F1 is the Gaussian hypergeometric function [41], and c0 is another constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Therefore our connection solution is such that its torsion and non-metricity read T λµν = −α 3 δλ0 µνc + p cos χ˜ϵ3λµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='48a) Qλµν = c � β 6 g(λµδ0 ν) + 18 − 12αα1 − βα2 √ 3 9(3α1 + α2 √ 3) � gλ(µδ0 ν) − gµνδ0 λ �� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='48b) with c given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Plugging this into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='37d), we find that −2r f ˇE00 = f′ + f r − k r + Λr + 2γ2p2 r3 + 2q2r γ1r4 + 8γp2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='49) – 24 – Since ˇE11 = −f−2 ˇE00, ˇE33 = sin2 χ ˇE22, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='50) and ˇE22 = r2 2f ˇE00 − � r3 2f ˇE00 �′ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='51) we only have to find the solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='49), which reads f = 1 − 2M r − Λr2 3 + 2γ2p2 r2 + 2q2 γ1r2 2F1 �1 4, 1, 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' −8γp2 γ1r4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='52) The symbol M stands for yet another integration constant, this time associated with the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The very interesting metric background (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='52) has been extensively studied in [40, 42], and there is no need to discuss it here in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In our case, nevertheless, the corrections to the Schwarzschild-(A)dS metric is due to a richer space-time geometry and not due to the introduction of additional fields (like a Maxwell field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In this sense, this is a novel result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Some comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Observe that by setting p = 0, the background (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='52) assumes the form f = 1 − 2M r + 8q2 r2 − Λr2 3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='53) and, up to choice of the integration constant q, it is indeed the metric solution in the MAGswell theory with action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='4) if we make the ansatz (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='43) for Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Moreover, the torsion and non-metricity of the connection solution, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='48), acquire the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='31), ergo, we recover the full solution in the MAGswell model, as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Another interesting setup is to consider the action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='35) with Λ = 0 = γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In this case, the connection solution will have torsion and non-metricity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='48) with c = c0 + qr3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='54) whereas the metric function f will be f = 1 − 2M r + 2γ2p2 r2 − Λeffr2 3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='55) where Λeff > 0 stands for the effective cosmological constant Λeff = 16γp2q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='56) Finally, non-singular solutions were reported in [40] for a specific choice of the mass parameter M in a strongly-coupled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The need to go to such a regime will not be necessary here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' we will just set γ2 = 0 and choose our mass parameter as M = M∗ := πq2 4(2γp2γ3 1)1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='57) Then, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='52) assumes the expression f = 1 − 2M∗ r − Λr2 3 + 2q2 γ1r2 2F1 �1 4, 1, 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' −8γp2 γ1r4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='58) – 25 – and admits the near-origin expansion f = r→0 1 − � q2 12γp2 + Λ 3 � r2 + O(r3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='59) If Λ ≥ 0 or −q2/(4γp2) < Λ < 0, the presence of a de Sitter core with radius ldS = 2p√3γ � q2 + 4γp2Λ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='60) is manifest, ensuring regularity of Riemann-curvature invariants at the origin and com- pleteness in the geodesic sense [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A further study of the causal structure of the solution reveals [40] that, for certain values (or ranges thereof) of the coupling/integration constants, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='58) describes either a gravitational soliton (horizon-free solution with regular origin), or just a standard black hole solution with an extremal limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' To the best of our knowledge, regular black hole solutions with torsion and non-metricity have not been yet reported in the MAG literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Since the actual novelty in the full solution is the existence of a non-trivial connection background, we find it worth to include a few lines about the behavior of the latter in various limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' First, let us write torsion and non-metricity in a coordinate-free manner by introducing a vierbein field eµ a, with indices a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' = (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', (3) and inverse ea µ satisfying the orthonormality relation gµν = ηabea µeb ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In particular, let us choose it to be diagonal, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', ea µ = diag �� f, 1 √f , r, r sin χ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='61) Then, the only non-vanishing components of T abc = ea λeµ b eν cT λµν are T (0) (1)(2) = T (1) (0)(2) = −T (2) (0)(1) = p cot χ r , T (i) (0)(i) = αc 3√f , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='62) where i, j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' take values in {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' It seems that the (i)(0)(i) components will be singular at the horizon radius r = r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Fortunately, this can be remedied by fixing the integration constants c0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='47) as c0 = − q γ1r+ 2F1 �1 4, 1, 5 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' −8γp2 γ1r4 + � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='63) so that c ∼ (r − r+) near the horizon surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Moreover, all components of the torsion tensor exhibit a r−1 fall-off at asymptotic infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Next, we have a single pole at the origin r = 0 due to the axial part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This pole persists even in the case of the regular metric solution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' If we assume that a probe particle with micro-structure follows the auto- parallels, then it is a good question to ask whether this particle is going to “feel” the torsion singularity at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Thankfully, the axial part of torsion drops out of the auto-parallel equation [20], and thus, this singular behavior should not really be a cause for concern!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, all components of Qabc = eλ aeµ b eν cQλµν are proportional to c/√f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' For f as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='52), this ratio vanishes at all previously discussed radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' On the other hand, in the case of the regular metric (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='58), it acquires a finite value at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the regular extremal case, it further is finite also at r = r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 26 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='3 Cosmological constant powered by torsion It is an old fact that the minimal coupling of a 3-form field to Einstein gravity without a cosmological constant leads to Einstein’s field equations with a cosmological constant purely derived from a gauge principle [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Here, we shall disclose a MAG model with no cosmological constant which also leads to pure gravity with a cosmological constant, the latter now powered by axial torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Let us consider the projective-invariant action IH = 1 2 � √−gd4x � R + 1 4H2 − 1 24F 2 (H) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='64) where F (H) λµνρ = 4∂[λHµνρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The purpose of the second term in the above integral is to cancel out the mass term for Hλµν present in the AF expression of the Ricci scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This ensures that the action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='64) is invariant under the translation H′λµν = Hλµν + ∂[λBµν] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='65) which corresponds to the transformation Γ′λ µν = Γλ µν − 1 2gλρ∂[ρBµν] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='66) of the affine connection in the FF, with Bµν being an arbitrary 2-form field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the convenient AF◦, the field equations read ˚ Oλµν1 = 1 6 ˜∇αF αλµν (H) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='67a) ˚ Oλµν 2 ≡ 1 2 � ˚tλµν − ˚q[µν]λ� = 0, ˚ Oλµν 3 ≡ −1 4˚πλµν = 0, ˚ Oλµν 4 ≡ ˚ O(µν)λ 2 + 1 8˚qλµν = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='67b) Bµ 1 ≡ −3 2Bµ 1 = 0, Bµ 2 ≡ 3 2Bµ 2 = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='67c) 2 ˆEµν = ˜Gµν − 1 2gµν � RT + RV + 1 4H2 − 1 24F 2 (H) � − 3 4˚πµαβ˚πναβ + +˚qαβµ˚qαβν + 1 2˚qµαβ˚qναβ +˚tαβµ˚tαβν + 1 2 ˚tµαβ˚tναβ +˚tαβ(µ˚qν) αβ − −˚qαβ (µ˚tν)αβ −˚tβα(µ˚qαβ ν) + 3 2BA µ BB ν ηAB − 1 6F (H) µ αβγF (H) ναβγ = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='67d) where the expressions of the BA µ ’s in terms of the AF (or FF) variables are to be found in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='7), RT in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) (first one), and RV in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' From the above, it is quite easy to conclude that ˚ OI = 0 = BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Therefore, the field equations assume the effective form ˜∇αF αλµν (H) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='68a) ˜Gµν = 1 48 � 8F (H) µ αβγF (H) ναβγ − gµνF 2 (H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='68b) To proceed, one must now recall that F (H) λρµν = χ(x)˜ϵλρµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='69) – 27 – since F(H) is a top-form in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Clearly, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='68a) implies that χ is an integration constant, say equal to χ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Consequently, we are left with ˜Gµν + 1 2χ2 0gµν = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='70) which will determine the metric, and we directly find that ˜Rµν = χ2 0 2 gµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='71) which is the familiar Riemannian Ricci-curvature condition for Einstein manifolds with positive constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As promised, we found a connection solution which features only axial torsion Hλµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Again, we stress out that this type of torsion has no effect on the auto-parallels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', the latter continue to coincide with the geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We also saw that our field equations do ef- fectively become Einstein’s field equations with a positive (effective) cosmological constant, Λeff = χ2 0/2, once we integrate out the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The cosmological solution in the ab- sence of matter sources would then be a de Sitter universe with Hubble constant H ∝ |χ0| where the expansion is now driven by an actual integration constant, powered by torsion, instead of an a priori fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 6 Summary and future prospects We started from the observation that the affine connection is a single field encoding n3-many off-shell DoF, arguing that, for certain purposes, it might be more efficient to distribute these degrees among more than one fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We then proceeded with a convenient change of field variables {g, Γ} → {g, ˚ ON, AI} going to a framework which we dubbed AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Besides the metric, the new field variables are the irreducible pieces of the torsion and non-metricity tensors under the Lorentz group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' They are thus automatically identified with the funda- mental fields {g, Γ} in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We worked out in detail the relations between the functional derivatives in the two frameworks and concluded that, not surprisingly, the field equations in the AF imply and are implied by the field equations in the FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Hence, the field equations in the AF constitute an equivalent system, and we have the freedom, at any stage, to switch between the different frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' To complete the mapping, we further disclosed a correspondence between linear connection transformations in the FF and translations in the AF while we also determined how the ˚ ON’s and the AI’s should transform under a local Weyl re-scaling of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We then applied the AF to the Hilbert-Palatini action and showed its well-known equiv- alence to Einstein gravity (up to choice of a gauge) also in the new framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Observing that a projective transformation of the connection corresponds to simultaneous translations of the AI’s in the AF, we further argued that the projective symmetry manifests itself as a true gauge symmetry in the new framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', one of the components of the vector triplet Aµ is redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In particular, this means that any projective-invariant action admits a description in terms of a reduced set of variables {g, ˚ ON, BA} where the BA’s are in general – 28 – identified with linear combinations of the AI’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This led us to develop a useful variant of the AF, which we dubbed diminished AF or AF◦ for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We saw that there exists a particular choice of combinations BA which reveals an SO(1, 1) symmetry of the n-dimensional HP action under a group action on the components of the doublet Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In n = 4, the field variables in the AF◦ can be re-organized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Using the fact that the dual of the 3-form torsion is a pseudo-vector, the quadruplet ˚ Oλµν is reduced to a triplet by handing its first component to the doublet Bµ which becomes a triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This is just a special four-dimensional variant of the AF◦, obtained via the change of variables {g, ˚ ON, BA} → {g, ˚ OI, BA}, which we called d-AF◦ for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As it turns out, the HP action proves to be an SO(1, 2)-symmetric action in the d-AF◦ where the group action mixes the BA’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Actions of the SO(2) subgroup rotate the elements of a two-dimensional subspace with the discrete version for θ = π/2 interpreted as a rotation of axial torsion to non-metricity, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Observing that any MAG theory in these alternative frameworks can be handled as a Riemannian theory with additional fields, we argued that it is an efficient strategy to use solvable (and suitable) Riemannian theories as “seeds” for solvable MAG theories which propagate the connection in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' As our first example, we drew inspiration from the elegant Einstein-Maxwell theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We proposed a theory for what we called the MAGswell field, a composite field labeling a projective-invariant linear combination of torsion and non-metricity traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The naive action should follow from the Maxwell action by replacing ˜R with R and the gauge field with the MAGswell field, the latter having nothing to do with a gauge connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Doing so, one of course notices that what was a U(1) of the second kind in the Riemannian case does not translate into a symmetry of the MAG theory under locally exact shifts of the MAGswell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The reason is that the presence of the Ricci scalar makes the field massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Thus, a counter-term Lagrangian was also included with the sole purpose of removing the mass terms for the constituents of the composite MAGswell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We then discussed the symmetries of the MAGswell action in all frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Exactly because the MAGswell field is a composite object, we showed that the action is symmetric under a 2-parameter transformation of the vector variables in the AF◦ (or the d-AF◦), which combines a transformation preserving the MAGswell field and one translating it by an exact vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the other frameworks, this symmetry shows up as a symmetry under a 3-parameter transformation, a fact attributed to the absorption of the projective-symmetry charge in the diminished AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We derived the field equations in the d-AF◦ and presented the solution in all frameworks, finding a proper expression that captures its form in all gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Actually, the reader was provided with table 1 which displays all cases possible, and which proves that the propagation of the MAGswell field, a gauge-independent fact, cannot be tied to a self-excitation of a uniquely determined part of the post-Riemannian structure in a gauge-independent fashion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', different parts of the connection background get excited for different choices of gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' After this instructive example, we proceeded with a more complicated theory, this time inspired by quasi-topological electromagnetism [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We proposed a Lagrangian with non- linear dynamics for the MAGswell field and the torsion pseudo-vector letting them interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' After briefly discussing the symmetries and deriving the field equations, – 29 – we adopted a static and spherically-symmetric metric ansatz, together with compatible ansätze for torsion and non-metricity, in an attempt to recover the black hole solution reported in [40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The full solution describes a black hole with a non-zero connection background sourcing the post-Schwarzschild contributions to the metric solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Under a certain tuning of the integration constants, we also showed that this black hole exhibits a regular core and is thus complete in the geodesic sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' However, assuming that particles with micro-structure follow auto-parallels, we also had to analyze the behavior of the torsion and non-metricity of the solution at all radii of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Doing so, we had to fix yet another integration constant to avoid a singular behavior at the horizon radius, but we concluded that there is no remedy for a single pole at the origin due to axial torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' This pole is inevitable even in the case of the regular black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Nevertheless, as we pointed out, the axial piece of torsion drops out from the auto- parallel equation meaning that the probe particle would never be affected by this torsion singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Finally, as our last example, and inspired by the derivation of a cosmological constant from a gauge principle [44], we put forth a simple MAG action for the 3-form torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' After deriving the field equations we presented a connection solution featuring only axial torsion which powers a positive effective cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The cosmological solution in this MAG theory — in the absence of matter sources — would be a de Sitter universe with the expansion driven by torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We remarked that the effective cosmological constant is an integration constant as opposed to a fixed-value Λ introduced by hand in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The main goal of this work was to communicate the idea that a smart change of field variables can be a really useful strategy when trying to find solvable MAG theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Indeed, our proposal proves to be a fruitful one, for although we restricted ourselves to showing only three examples, these are suggestive of many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Writing down simple field theories for the new field variables compared to considering combinations of curvature invariants to make the connection dynamical, is of course a far less general method, albeit a much more targeted and result-oriented one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' In the future, we plan to give more examples and solutions which are not necessarily inspired by Riemannian theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' We find it interesting to study kinetic theories for the various tensor modes and also investigate if (and how) Riemannian theories with scalar fields can fit as an inspiration into this description of MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' A Irreducible decomposition of a rank-3 tensor The irreducible decomposition of a general rank-3 tensor ∆λµν under the Lorentz group reads ∆λµν = ∆[λµν] + ˚∆(λµν) + ˚ Dλ[µν] + ˚ Dλ(µν) + ¯∆(λµν) + ¯Dλ[µν] + ¯Dλ(µν), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='1) – 30 – where ˚ Dλµν = ∆λµν − ∆(λµν) − ∆[λµν] − 1 n − 1gλ[µ � ∆α |α|ν] − ∆α ν]α � − − 1 3(n − 1)gλ(µ � ∆α |α|ν) + ∆α ν)α − 2∆ν)α α� + + 1 3(n − 1)gµν (∆ααλ + ∆αλα − 2∆λαα) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2a) ˚∆(λµν) = ∆(λµν) − 1 D + 2g(µν � ∆λ)α α + ∆α λ)α + ∆α |α|λ) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2b) ¯∆(λµν) = 1 n + 2g(µν � ∆λ)α α + ∆α λ)α + ∆α |α|λ) � = ∆(λµν) − ˚∆(λµν), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2c) ¯Dλµν = 1 n − 1gλ[µ � ∆α |α|ν] − ∆α ν]α � − 1 3(n − 1)gµν (∆ααλ + ∆αλα − 2∆λαα) + + 1 3(n − 1)gλ(µ � ∆α |α|ν) + ∆α ν)α − 2∆ν)α α� = Dλµν − ˚ Dλµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='2d) B Glossary Indices Values µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=',n − 1 i, j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=',n − 1 a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' (0),(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=',(n − 1) M, N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 1, 2, 3, 4 I, J, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 2, 3, 4 A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 1, 2 A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' 1, 2, 3 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Indices used in this work and their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Acronym Full name MAG Metric-affine gravity DoF Degrees of freedom FF Fundamental framework, {g, Γ} AF Alternative framework, {g, ˚ ON, AI} AF◦ Diminished alternative framework, {g, ˚ ON, BA} d-AF◦ , {g, ˚ OI, BA} Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Acronyms used in this work and their full name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 31 – Symbol Definition ˚ Oλµν {Hλµν,˚tλµν,˚πλµν,˚qλµν} Aµ {Tµ, ρµ, uµ} Bµ in the AF◦ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='21) Bµ in the d-AF◦ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='43) g det(gµν) ˚ Oλµν N √−g−1δI/δ˚ ON λµν Aµ I √−g−1δI/δAI µ Bµ A/A √−g−1δI/δBA/A µ ∆λµν √−g−1δI/δΓλ µν Eµν √−g−1δI/δgµν in the FF ˆEµν √−g−1δI/δgµν in the AF ˇEµν √−g−1δI/δgµν in the d-AF◦ RT , RV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='9) RV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='22) ˆRT , ˆRV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='42) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Some symbols used in this work and their definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Acknowledgments D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' work is funded by the Estonian Research Council grant (SJD14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' acknowledges financial support provided by the European Regional Development Fund (ERDF) through the Center of Excellence TK133 “The Dark Side of the Universe” and PRG356 “Gauge gravity: unification, extensions and phenomenology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' also acknowledges participation in the COST Association Action CA18108 “Quantum Gravity Phenomenology in the Mul- timessenger Approach (QG-MM)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' The authors would also like to thank Anastasios C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Petkou for the fruitful discussions and valuable comments during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' References [1] Jerome Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Comptes Rendus Physique, 13:566–665, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' [2] Charles W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Misner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Mixmaster universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=', 22:1071–1074, May 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Dicke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Gravitation and the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' American Philosophical Society: Memoirs of the American Philosophical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' American Philosophical Society, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Dicke and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Peebles.' 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+page_content=' van Nieuwenhuizen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Quantum Inequivalence of Different Field Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' B, 94:179–182, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} +page_content=' – 34 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFIT4oBgHgl3EQfzSt7/content/2301.11364v1.pdf'} diff --git a/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf 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V. DEDANIA +Dept. of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388120, +Gujarat, India +J. G. PATEL* +Dept. of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388120, +Gujarat, India +Abstract. We prove that the faithfull and uniqueness of norm properties +are stable in different product algebras such as direct-sum product algebra, +convolution product algebra, and module product algebra. Further, we exhibit +that these properties are not stable in null product algebra, and also give a +common sufficient condition in terms of algebra norm for the co-dimension of +A2 = span{ab : a, b ∈ A} to be finite in A and A2 = A (when A2 = A). +1. Introduction +Throughout A is an (associative) algebra over the complex field C. A norm ∥ · ∥ +on A, means it is a linear norm and submultiplicative, i.e. ∥ab∥ ≤ ∥a∥∥b∥ (a, b ∈ A). +If ∥ · ∥ gives a complete metric topology, then we say A is a Banach algebra. The +sets N(A) and Nc(A) denote the set of all algebra norms (up to equivalent), the set +of all Banach algebra norms (up to equivalent), respectively. Let I be an ideal in +an algebra A. Then A is faithful over I if ab = 0 (b ∈ I), then a = 0. An algebra +A is faithful if it is faithful over itself [Da:00]. It is shown in [DP:22(a)] that the +uniqueness of norm property on the cartesian product algebra A × B depends on +the uniqueness of norm on A and B, where A and B are Banach algebras. So, +it is natural to study the uniqueness of norm on other product algebras namely, +the direct-sum product algebra [Definition 2.2], the convolution product algebra +[Definition 2.6] and the module product algebra [Definition 2.11]. In some cases, we +obtained necessary and sufficient conditions for uniqueness of norm and faithfulness. +Dales and Loy have supplied an interesting example that it has two inequivalent +complete norms [DL:97]. We proved in [DP:22(a)], [DP:22(b)] that it has infinitely +many complete norms and incomplete norms. In this paper, we generalised this +result for a normed algebra A with some condition on A. We also note down that +E-mail addresses: hvdedania@gmail.com, jatinprofessor39@gmail.com. +2020 Mathematics Subject Classification. 46H05, 46H25. +Key words and phrases. Algebra, Banach Algebra, Faithful, Norm . +1 + +2 +H. V. DEDANIA AND J. G. PATEL* +the condition “N(A) = Nc(A) contains only one element” served as a sufficient +condition for the co-dimension of A2 to be finite in A, and A2 = A (in the case of +A2 is dense in A). +2. Main Results +Proposition 2.1. [Wa:14, P. 14] Let A be a Banach algebra, and let I1, I2 be +closed ideals in A with I1 ⊂ I2. Suppose that both A/I2 and I2/I1 have unique +algebra norms. Then A/I1 has a unique algebra norm. +Definition 2.2. [Ka:16] Let A be an algebra and B be a subalgebra of A. Then +A × B is an algebra with pointwise linear operations and direct-sum product ‘×d’ +define as (a, b) ×d (c, d) = (ac + ad + bc, bd) ((a, b), (c, d) ∈ A ×d B). The direct-sum +product is a generalization of the product defined on Ae (means unitization of A). +Further, ∥(a, b)∥1 = ∥a∥ + ∥b∥ and |(a, b)| = max{∥a − b∥, ∥b∥} ((a, b) ∈ A ×d B) +are algebra norms on A ×d B. +Remark 2.3. (i) Let (A, ∥ · ∥) be a Banach algebra and B be a closed subalgebra of +A. Then (A ×d B, ∥ · ∥1) is a Banach algebra. +(ii) A × {0} is a closed ideal of A ×d B. +(iii) By the first isomorphism theorem, (A ×d B)/A ∼= B. +(iv) ∥ · ∥1 and | · | are equivalent on A ×d B. +Lemma 2.4. Let A be an algebra and B be a subalgebra of A. Then A and B are +faithful if and only if A ×d B is faithful. +Proof. Assume that A and B are faithful. Let (a, b) ∈ A×d B with (a, b)×d (c, d) = +(0, 0) ((c, d) ∈ A ×d B). Then (ac + ad + bc, bd) = (0, 0) ((c, d) ∈ A ×d B). That +implies b = 0 as B is faithful. Put b = 0 in ac + ad + bc = 0, then a(c + d) = 0 as +d was an arbitrary element, we can put d = 0. That gives a = 0 as A is faithful. +Therefore A×d B is faithful. Conversely, assume that A×d B is faithful. Suppose, if +possible that there exists 0 ̸= a ∈ A such that ax = 0 (x ∈ A). Set (a, 0) ∈ A ×d B +and let (x, y) ∈ A ×d B. Then (a, 0) ×d (x, y) = (ax + ay + 0x, 0y) = (0, 0), which +gives contradiction to faithfulness of A ×d B. There fore A is faithful. Similarly, +suppose B is not faithful, there exists 0 ̸= b ∈ B such that by = 0 (y ∈ B). Set +(−b, b) ∈ A×dB and let (x, y) ∈ A×dB. Then (−b, b)×d(x, y) = (−bx−by+bx, by) = +(−by, by) = (0, 0), which gives a contradiction to faithfulness of A ×d B. Hence B +is faithful. +□ +Theorem 2.5. Let (A, ∥ · ∥) be a Banach algebra and B be a closed subalgebra of +A. Then N(A) and N(B) are singleton if and only if N(A ×d B) is singleton. +Proof. Assume that N(A) and N(B) are singleton. Consider J1 = {(0, 0)} and +J2 = A×{0}. Then J1 and J2 are closed ideals in A×dB, J1 ⊂ J2, (A×dB)/J2 ∼= B, +and J2/J1 ∼= A. Since N(A) and N(B) are singleton, N((A×dB)/J2) and N(J2/J1) +are singleton. Hence, by Proposition 2.1, A ×d B has the unique algebra norm, i.e., +N(A ×d B) is singleton. Conversely, assume that N(A × B) is singleton. Suppose, +if possible, N(A) is not singleton. +Choose two norms p1(·), p2(·) ∈ N(A) such +that �p1(·) and �p2(·) are two different classes in N(A). Then p1(·) and p2(·) are +not equivalent on A. Next take q(·) ∈ N(B). Define q1(x, y) = p1(x) + q(y) and +q2(x, y) = p2(x) + q(y) for (x, y) ∈ A × B. Then q1(·) and q2(·) are non-equivalent +norms on A × B, and hence N(A × B) is not singleton, which is a contradiction. +Thus N(A) must be singleton. Similarly, N(B) is singleton. +□ + +H. V. DEDANIA AND J. G. PATEL* +3 +Definition 2.6. [Ka:16] Let A be an algebra and I be an ideal of A. Then A × I +is an algebra with co-ordinatewise linear operations and the convolution product +‘×c’ define as (a, x) ×c (b, y) = (ab + xy, ay + xb) ((a, x), (b, y) ∈ A ×c I). Then +A ×c I is called a convolution product algebra. +Remark 2.7. (i) Let I be a closed ideal in A. Then I ×c I is a closed ideal in +A ×c I. +(ii) By the first isomorphism theorem, (A ×c I)/(I ×c I) ∼= A/I. +Lemma 2.8. Let A be an algebra and I be an ideal in A. Then A is faithful if +and only if A ×c I is faithful. +Proof. Assume that A is faithful. +Let (a, x) ∈ A ×c I with (a, x) ×c (b, x) = +(0, 0) ((b, x) ∈ A ×c I). Then (a, x) ×c (b, 0) = (ab, xb) = (0, 0) ((b, 0) ∈ A ×c I), +ab = xb = 0 (b ∈ A). Hence a = x = 0 as A is faithful. Conversely, assume that +A ×c I is faithful. Suppose, if possible, that there exists 0 ̸= a ∈ A such that +ax = 0 (x ∈ A). Set (a, 0) ∈ A ×c I and let (x, y) ∈ A ×c I. Then (a, 0) ×c (b, y) = +(ab, ay) = (0, 0), which gives a contradiction to faithfulness of A ×c I. Hence A is +faithful. +□ +Lemma 2.9. Let A be an algebra and I be a closed ideal of A. Then N(A ×c I) +is singleton implies N(A) is singleton. +Proof. The proof is similar to the “converse part” of proof of Theorem 2.5. +□ +Theorem 2.10. Let A be a Banach algebra and I be a closed ideal of A. Then +N(A/I) and N(I ×c I) are singleton implies N(A ×c I) is singleton. +Proof. Set J1 = {0} × {0} and J2 = I × I. Then J1 and J2 are two closed ideals of +A, (A×cI/J2) ∼= A/I and J2/J1 ∼= J2. Since N(A/I) and N(I ×cI) are singleton, +N(A ×c I/J2) and N(J2/J1) are singleton. Hence, by Proposition 2.1, N(A ×c I) +is singleton. +□ +Definition 2.11. [Da:00] Let (A, ∥·∥) be a Banach algebra and (X, |·|) be a Banach +A-bimodule. For (a, x), (b, y) ∈ A × X, define (a, x) ×m (b, y) = (ab, ay + xb). Then +(A ×m X, ×m) is called the module product algebra. It is a Banach algebra with +the norm ∥(a, x)∥1 = ∥a∥ + |x| ((a, x) ∈ A ×m X). +Remark 2.12. (i) {0} × X is a closed ideal in A ×m X. +(ii) By the first isomorphism theorem, (A ×m X)/({0} × X) ∼= A. +Definition 2.13. Let X be an A-bimodule. It is said to be A-faithful if xa = +0 (a ∈ A), then x = 0. +Lemma 2.14. If A is faithful and X is A-faithful, then A ×m X is faithful. +Proof. Let (a, x) ∈ A ×m X such that (a, x) ×m (b, y) = (0, 0) ((b, y) ∈ A ×m X). +Then (a, x) ×m (b, 0) = (ab, xb) = (0, 0) (b ∈ A). Hence a = 0 as A is faithful and +x = 0 as X is A-faithful. +□ +Theorem 2.15. Let (A, ∥·∥) be a Banach algebra and X be a Banach A-bimodule. +Then N(A ×m X) is singleton if and only if N(A) and N(X) are singleton. +Proof. Assume that N(A) and N(X) are singleton. Set J1 = {(0, 0)} and J2 = +{0} × X. Then J1 and J2 are closed ideals in A ×m X, J1 ⊂ J2, (A ×m X)/J2 ∼= A, +and J2/J1 ∼= X. +Since N(A) and N(X) are singleton, N((A ×m X)/J2) and + +4 +H. V. DEDANIA AND J. G. PATEL* +N(J2/J1) are singleton. Hence, by Proposition 2.1, A ×m X has unique algebra +norm, i.e., N(A ×m X) is singleton. The proof of “converse part” is similar to the +proof of Theorem 2.5. +□ +Definition 2.16. Let (A, ∥ · ∥) be a normed algebra, and set A ×0 C, with product +(a, α)×0(b, β) = ab and ∥(a, α)∥ = ∥a∥+|α| ((a, α) ∈ A×0C). Then (A×0C, ×0, ∥·∥) +is a normed algebra with rad(A ×0 C) = C. +The next result is a generalization of given example in [DL:97, P. 633]. +Theorem 2.17. Let A be a normed algebra with the co-dimension of A2 being +infinite. +Then N(A ×0 C) is infinite. +Moreover, if Nc(A) is non-empty, then +Nc(A ×0 C) is infinite. +Proof. Let (A, ∥ · ∥) be a normed algebra. Since A2 has infinite co-dimension in A, +there exist infinite linearly independent subset L of A such that A2 ∩ L = φ and +∥a∥ = 1 (a ∈ L). For each n ∈ N, choose Ln = {an1, an2, . . .} ⊂ L such that: +(1) Each Ln is infinite; +(2) Ln ∩ Lm = φ (n ̸= m); +(3) L = �∞ +n=1 Ln. +Let Bn be a (Hamel) basis of A such that Ln ⊂ Bn for each n ∈ N. Then each +Cn = Bn ∩ A2 is a basis of A2. Consider the (unique) linear map ϕn : A −→ C +such that +ϕn(a) = + + + + + +k +(if a = ank ∈ Ln \ {an1}); +1 +(if a ∈ Bn \ (Ln ∪ Cn)); +0 +(if a ∈ Cn ∪ {an1}). +Next take B = A ×0 C. For (a, α), (b, β) ∈ B, define +(a, α)(b, β) = (ab, 0) +and +p((a, α)) = ∥a∥ + |α|. +Then (B, p(·)) is a normed algebra. For each n ∈ N, define +pn((a, α)) = ∥a∥ + |ϕn(a) − α| +((a, α) ∈ A). +Clearly, each pn(·) is a linear norm. Let (a, α), (b, β) ∈ B. Then pn((a, α)(b, β)) = +pn((ab, 0)) = ∥ab∥ ≤ pn((a, α))pn((b, β)) because ab ∈ A2 and hence ϕn(ab) = 0. +Thus each pn(·) ∈ N(B). +Now, we claim that these norms are non-equivalent. +Let m < n and gk = amk (k ∈ N). Then, for k ≥ 2, pm((ak, 0)) = 1 + k and +pn((ak, 0)) ≤ 2 because ϕn(amk) = 0 or 1. Thus we have proved our claim. Hence +Nc(A) is an infinite set. Suppose (A, ∥ · ∥) is a Banach algebra. Finally, we claim +that each pn(·) ∈ Nc(B). Let (an, αn) be a Cauchy sequence in (B, pn(·)). Then an +is a Cauchy sequence in (A, ∥ · ∥), converges to a and αn is a Cauchy sequence in +C, converges to α. Hence (an, αn) converges to (a, α) ∈ B. +□ +The next result is a direct application of Proposition 2.1 and providing sufficient +condition for the co-dimension of A2 is finite in A. +Corollary 2.18. Let A be an algebra such that N(A) and Nc(A) are singleton. +Then +(i) N(A ×0 C) is singleton. +(ii) the co-dimension of A2 is finite in A. +(iii) if A2 is dense in A, then A2 = A. + +H. V. DEDANIA AND J. G. PATEL* +5 +Proof. (i) Since (A ×0 C)/{0} ×0 C ∼= A and {0} ×0 C/{0} ×0 {0} have unique +algebra norms. Hence by Proposition 2.1, (A ×0 C)/{0} ×0 {0} ∼= A ×0 C has a +unique algebra norm. +(ii) Suppose if possible the co-dimension of A2 is infinite in A. Then by above +Theorem 2.17, the Banach algebra (A ×0 C, p(·)) has infinitely many norms, but +by Statement (i), the Banach algebra (A ×0 C, p(·)) has a unique norm. Hence the +co-dimension of A2 is finite in A. +(iii) Suppose the co-dimension of A2 is n in A for some n ∈ N. Then the set +B = {v1 + A2, v2 + A2 . . . , vn + A2} is a basis of A/A2. +Now consider W = +span{A2, v1, v2, . . . , vn−1} is a hyperspace of A, i.e. dim(A/W) = 1. Then there +exist discontinuous linear functional ϕ : A −→ C such that kerϕ = W. Define +∥|a∥| = ∥a∥ + |ϕ(a)| (a ∈ A) and ∥a∥ ≤ ∥|a∥| (a ∈ A). Hence n must be 0 and +that gives A2 = A. +□ +3. Examples +Example 3.1. Let A = M2(C) and ideal I = +�� +a +0 +0 +0 +� +, +� +0 +b +0 +0 +�� +. Then I is a +right non-faithful ideal of a faithful algebra M2(C). +Example 3.2. Let 1 ≤ p < ∞ and let A = ℓp with pointwise product. Then +N(ℓp) > 1 and (ℓp)2 = ℓ +p +2 is dense in (ℓp, ∥ · ∥p) but ℓ +p +2 ⊊ ℓp. This example says +that we can not relaxe the condition “N(A) is singleton” from Corollary 2.18(iii). +Example 3.3. Every ideal of a commutative semisimple algebra is faithful. +4. Question +It would be interesting to examine whether Proposition 2.1 hold without assuming +completeness. +Declarations +Funding The second author is very thankful to the University Grants Commission +(UGC), New Delhi, for providing Senior Research Fellowship. +Conflict of interest +The authors have no relevant financial or non-financial +interests to disclose. +References +[Da:00] +H. G. Dales, Banach algebras and automatic continuity, Oxford Science Publication, +London Math. Soc. Monographs, 2000. +[DL:97] +H. G. Dales and R. J. Loy, Uniqueness of the norm topology for Banach algebras with +finite-dimensional radical, Proc. London Math. Soc., 74(3) (1997) 633-661. +[DP:22(a)] H. V. Dedania and J. G. Patel, Classification of algebra norms and relations among +them, Rend. Circ. Mat. Palermo(2), Accepted. +[DP:22(b)] H. V. Dedania and J. G. Patel, Methods for constructing algebra norms on Banach +algebras, Bull. Calcutta Math. Soc., 114(3) 2022, 587-596. +[Ka:16] +H. J. Kanani, Spectral and uniqueness properties in various Banach algebra products, +Ph.D. Thesis, Sardar Patel University, 2016. +[Wa:14] +G. K. Ware, Uniqueness of norm properties of Calkin algebras, Ph.D. Thesis, Aus- +tralian National University, 2014. + diff --git a/ydE2T4oBgHgl3EQfhgdF/content/tmp_files/load_file.txt b/ydE2T4oBgHgl3EQfhgdF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..48752358e091f70e3757fe130fcc1e5bb11a3adc --- /dev/null +++ b/ydE2T4oBgHgl3EQfhgdF/content/tmp_files/load_file.txt @@ -0,0 +1,326 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf,len=325 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='03948v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='FA] 10 Jan 2023 UNIQUENESS OF NORM AND FAITHFULNESS OF SOME PRODUCT BANACH ALGEBRAS H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' DEDANIA Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388120, Gujarat, India J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' PATEL* Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388120, Gujarat, India Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' We prove that the faithfull and uniqueness of norm properties are stable in different product algebras such as direct-sum product algebra, convolution product algebra, and module product algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Further, we exhibit that these properties are not stable in null product algebra, and also give a common sufficient condition in terms of algebra norm for the co-dimension of A2 = span{ab : a, b ∈ A} to be finite in A and A2 = A (when A2 = A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Introduction Throughout A is an (associative) algebra over the complex field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' A norm ∥ · ∥ on A, means it is a linear norm and submultiplicative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' ∥ab∥ ≤ ∥a∥∥b∥ (a, b ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' If ∥ · ∥ gives a complete metric topology, then we say A is a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' The sets N(A) and Nc(A) denote the set of all algebra norms (up to equivalent), the set of all Banach algebra norms (up to equivalent), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let I be an ideal in an algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A is faithful over I if ab = 0 (b ∈ I), then a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' An algebra A is faithful if it is faithful over itself [Da:00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' It is shown in [DP:22(a)] that the uniqueness of norm property on the cartesian product algebra A × B depends on the uniqueness of norm on A and B, where A and B are Banach algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' So, it is natural to study the uniqueness of norm on other product algebras namely, the direct-sum product algebra [Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='2], the convolution product algebra [Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='6] and the module product algebra [Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' In some cases, we obtained necessary and sufficient conditions for uniqueness of norm and faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Dales and Loy have supplied an interesting example that it has two inequivalent complete norms [DL:97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' We proved in [DP:22(a)], [DP:22(b)] that it has infinitely many complete norms and incomplete norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' In this paper, we generalised this result for a normed algebra A with some condition on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' We also note down that E-mail addresses: hvdedania@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='com, jatinprofessor39@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 46H05, 46H25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Algebra, Banach Algebra, Faithful, Norm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 1 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' DEDANIA AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' PATEL* the condition “N(A) = Nc(A) contains only one element” served as a sufficient condition for the co-dimension of A2 to be finite in A, and A2 = A (in the case of A2 is dense in A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Main Results Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' [Wa:14, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 14] Let A be a Banach algebra, and let I1, I2 be closed ideals in A with I1 ⊂ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Suppose that both A/I2 and I2/I1 have unique algebra norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A/I1 has a unique algebra norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' [Ka:16] Let A be an algebra and B be a subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A × B is an algebra with pointwise linear operations and direct-sum product ‘×d’ define as (a, b) ×d (c, d) = (ac + ad + bc, bd) ((a, b), (c, d) ∈ A ×d B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' The direct-sum product is a generalization of the product defined on Ae (means unitization of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Further, ∥(a, b)∥1 = ∥a∥ + ∥b∥ and |(a, b)| = max{∥a − b∥, ∥b∥} ((a, b) ∈ A ×d B) are algebra norms on A ×d B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (i) Let (A, ∥ · ∥) be a Banach algebra and B be a closed subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (A ×d B, ∥ · ∥1) is a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (ii) A × {0} is a closed ideal of A ×d B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (iii) By the first isomorphism theorem, (A ×d B)/A ∼= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (iv) ∥ · ∥1 and | · | are equivalent on A ×d B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A be an algebra and B be a subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A and B are faithful if and only if A ×d B is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Assume that A and B are faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (a, b) ∈ A×d B with (a, b)×d (c, d) = (0, 0) ((c, d) ∈ A ×d B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (ac + ad + bc, bd) = (0, 0) ((c, d) ∈ A ×d B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' That implies b = 0 as B is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Put b = 0 in ac + ad + bc = 0, then a(c + d) = 0 as d was an arbitrary element, we can put d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' That gives a = 0 as A is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Therefore A×d B is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Conversely, assume that A×d B is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Suppose, if possible that there exists 0 ̸= a ∈ A such that ax = 0 (x ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Set (a, 0) ∈ A ×d B and let (x, y) ∈ A ×d B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (a, 0) ×d (x, y) = (ax + ay + 0x, 0y) = (0, 0), which gives contradiction to faithfulness of A ×d B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' There fore A is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Similarly, suppose B is not faithful, there exists 0 ̸= b ∈ B such that by = 0 (y ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Set (−b, b) ∈ A×dB and let (x, y) ∈ A×dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (−b, b)×d(x, y) = (−bx−by+bx, by) = (−by, by) = (0, 0), which gives a contradiction to faithfulness of A ×d B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence B is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (A, ∥ · ∥) be a Banach algebra and B be a closed subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then N(A) and N(B) are singleton if and only if N(A ×d B) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Assume that N(A) and N(B) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Consider J1 = {(0, 0)} and J2 = A×{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then J1 and J2 are closed ideals in A×dB, J1 ⊂ J2, (A×dB)/J2 ∼= B, and J2/J1 ∼= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Since N(A) and N(B) are singleton, N((A×dB)/J2) and N(J2/J1) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1, A ×d B has the unique algebra norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=', N(A ×d B) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Conversely, assume that N(A × B) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Suppose, if possible, N(A) is not singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Choose two norms p1(·), p2(·) ∈ N(A) such that �p1(·) and �p2(·) are two different classes in N(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then p1(·) and p2(·) are not equivalent on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Next take q(·) ∈ N(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Define q1(x, y) = p1(x) + q(y) and q2(x, y) = p2(x) + q(y) for (x, y) ∈ A × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then q1(·) and q2(·) are non-equivalent norms on A × B, and hence N(A × B) is not singleton, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Thus N(A) must be singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Similarly, N(B) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' DEDANIA AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' PATEL* 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' [Ka:16] Let A be an algebra and I be an ideal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A × I is an algebra with co-ordinatewise linear operations and the convolution product ‘×c’ define as (a, x) ×c (b, y) = (ab + xy, ay + xb) ((a, x), (b, y) ∈ A ×c I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A ×c I is called a convolution product algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (i) Let I be a closed ideal in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then I ×c I is a closed ideal in A ×c I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (ii) By the first isomorphism theorem, (A ×c I)/(I ×c I) ∼= A/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A be an algebra and I be an ideal in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then A is faithful if and only if A ×c I is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Assume that A is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (a, x) ∈ A ×c I with (a, x) ×c (b, x) = (0, 0) ((b, x) ∈ A ×c I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (a, x) ×c (b, 0) = (ab, xb) = (0, 0) ((b, 0) ∈ A ×c I), ab = xb = 0 (b ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence a = x = 0 as A is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Conversely, assume that A ×c I is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Suppose, if possible, that there exists 0 ̸= a ∈ A such that ax = 0 (x ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Set (a, 0) ∈ A ×c I and let (x, y) ∈ A ×c I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (a, 0) ×c (b, y) = (ab, ay) = (0, 0), which gives a contradiction to faithfulness of A ×c I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence A is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A be an algebra and I be a closed ideal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then N(A ×c I) is singleton implies N(A) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' The proof is similar to the “converse part” of proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A be a Banach algebra and I be a closed ideal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then N(A/I) and N(I ×c I) are singleton implies N(A ×c I) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Set J1 = {0} × {0} and J2 = I × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then J1 and J2 are two closed ideals of A, (A×cI/J2) ∼= A/I and J2/J1 ∼= J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Since N(A/I) and N(I ×cI) are singleton, N(A ×c I/J2) and N(J2/J1) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1, N(A ×c I) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' [Da:00] Let (A, ∥·∥) be a Banach algebra and (X, |·|) be a Banach A-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' For (a, x), (b, y) ∈ A × X, define (a, x) ×m (b, y) = (ab, ay + xb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (A ×m X, ×m) is called the module product algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' It is a Banach algebra with the norm ∥(a, x)∥1 = ∥a∥ + |x| ((a, x) ∈ A ×m X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (i) {0} × X is a closed ideal in A ×m X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (ii) By the first isomorphism theorem, (A ×m X)/({0} × X) ∼= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let X be an A-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' It is said to be A-faithful if xa = 0 (a ∈ A), then x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' If A is faithful and X is A-faithful, then A ×m X is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (a, x) ∈ A ×m X such that (a, x) ×m (b, y) = (0, 0) ((b, y) ∈ A ×m X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (a, x) ×m (b, 0) = (ab, xb) = (0, 0) (b ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence a = 0 as A is faithful and x = 0 as X is A-faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (A, ∥·∥) be a Banach algebra and X be a Banach A-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then N(A ×m X) is singleton if and only if N(A) and N(X) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Assume that N(A) and N(X) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Set J1 = {(0, 0)} and J2 = {0} × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then J1 and J2 are closed ideals in A ×m X, J1 ⊂ J2, (A ×m X)/J2 ∼= A, and J2/J1 ∼= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Since N(A) and N(X) are singleton, N((A ×m X)/J2) and 4 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' DEDANIA AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' PATEL* N(J2/J1) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1, A ×m X has unique algebra norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=', N(A ×m X) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' The proof of “converse part” is similar to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (A, ∥ · ∥) be a normed algebra, and set A ×0 C, with product (a, α)×0(b, β) = ab and ∥(a, α)∥ = ∥a∥+|α| ((a, α) ∈ A×0C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (A×0C, ×0, ∥·∥) is a normed algebra with rad(A ×0 C) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' The next result is a generalization of given example in [DL:97, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 633].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A be a normed algebra with the co-dimension of A2 being infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then N(A ×0 C) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Moreover, if Nc(A) is non-empty, then Nc(A ×0 C) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (A, ∥ · ∥) be a normed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Since A2 has infinite co-dimension in A, there exist infinite linearly independent subset L of A such that A2 ∩ L = φ and ∥a∥ = 1 (a ∈ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' For each n ∈ N, choose Ln = {an1, an2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='} ⊂ L such that: (1) Each Ln is infinite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (2) Ln ∩ Lm = φ (n ̸= m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (3) L = �∞ n=1 Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let Bn be a (Hamel) basis of A such that Ln ⊂ Bn for each n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then each Cn = Bn ∩ A2 is a basis of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Consider the (unique) linear map ϕn : A −→ C such that ϕn(a) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 k (if a = ank ∈ Ln \\ {an1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 1 (if a ∈ Bn \\ (Ln ∪ Cn));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 0 (if a ∈ Cn ∪ {an1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Next take B = A ×0 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' For (a, α), (b, β) ∈ B, define (a, α)(b, β) = (ab, 0) and p((a, α)) = ∥a∥ + |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (B, p(·)) is a normed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' For each n ∈ N, define pn((a, α)) = ∥a∥ + |ϕn(a) − α| ((a, α) ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Clearly, each pn(·) is a linear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (a, α), (b, β) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then pn((a, α)(b, β)) = pn((ab, 0)) = ∥ab∥ ≤ pn((a, α))pn((b, β)) because ab ∈ A2 and hence ϕn(ab) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Thus each pn(·) ∈ N(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Now, we claim that these norms are non-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let m < n and gk = amk (k ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then, for k ≥ 2, pm((ak, 0)) = 1 + k and pn((ak, 0)) ≤ 2 because ϕn(amk) = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Thus we have proved our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence Nc(A) is an infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Suppose (A, ∥ · ∥) is a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Finally, we claim that each pn(·) ∈ Nc(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let (an, αn) be a Cauchy sequence in (B, pn(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then an is a Cauchy sequence in (A, ∥ · ∥), converges to a and αn is a Cauchy sequence in C, converges to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence (an, αn) converges to (a, α) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ The next result is a direct application of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1 and providing sufficient condition for the co-dimension of A2 is finite in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A be an algebra such that N(A) and Nc(A) are singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then (i) N(A ×0 C) is singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (ii) the co-dimension of A2 is finite in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (iii) if A2 is dense in A, then A2 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' DEDANIA AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' PATEL* 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (i) Since (A ×0 C)/{0} ×0 C ∼= A and {0} ×0 C/{0} ×0 {0} have unique algebra norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1, (A ×0 C)/{0} ×0 {0} ∼= A ×0 C has a unique algebra norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (ii) Suppose if possible the co-dimension of A2 is infinite in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then by above Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='17, the Banach algebra (A ×0 C, p(·)) has infinitely many norms, but by Statement (i), the Banach algebra (A ×0 C, p(·)) has a unique norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence the co-dimension of A2 is finite in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' (iii) Suppose the co-dimension of A2 is n in A for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then the set B = {v1 + A2, v2 + A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' , vn + A2} is a basis of A/A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Now consider W = span{A2, v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' , vn−1} is a hyperspace of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' dim(A/W) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then there exist discontinuous linear functional ϕ : A −→ C such that kerϕ = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Define ∥|a∥| = ∥a∥ + |ϕ(a)| (a ∈ A) and ∥a∥ ≤ ∥|a∥| (a ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Hence n must be 0 and that gives A2 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Examples Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let A = M2(C) and ideal I = �� a 0 0 0 � , � 0 b 0 0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then I is a right non-faithful ideal of a faithful algebra M2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Let 1 ≤ p < ∞ and let A = ℓp with pointwise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Then N(ℓp) > 1 and (ℓp)2 = ℓ p 2 is dense in (ℓp, ∥ · ∥p) but ℓ p 2 ⊊ ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' This example says that we can not relaxe the condition “N(A) is singleton” from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='18(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Every ideal of a commutative semisimple algebra is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Question It would be interesting to examine whether Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='1 hold without assuming completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Declarations Funding The second author is very thankful to the University Grants Commission (UGC), New Delhi, for providing Senior Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Conflict of interest The authors have no relevant financial or non-financial interests to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' References [Da:00] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Dales, Banach algebras and automatic continuity, Oxford Science Publication, London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Monographs, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' [DL:97] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Dales and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Loy, Uniqueness of the norm topology for Banach algebras with finite-dimensional radical, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Soc.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} +page_content=' Thesis, Aus- tralian National University, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQfhgdF/content/2301.03948v1.pdf'} diff --git a/zNAzT4oBgHgl3EQfC_oX/vector_store/index.faiss b/zNAzT4oBgHgl3EQfC_oX/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9fdb1545efc8a29f5695465a4ce50932d77f0a88 --- /dev/null +++ b/zNAzT4oBgHgl3EQfC_oX/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3c996143287a9ef10a5e1c1d25aaed6698414599c26cb82c419d77fdffd481e6 +size 8126509 diff --git a/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf b/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e253eabc2840976384c5afd297ba4bde5cb3668b Binary files /dev/null and b/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf differ diff --git a/zdE4T4oBgHgl3EQfZAyd/content/tmp_files/2301.05053v1.pdf.txt b/zdE4T4oBgHgl3EQfZAyd/content/tmp_files/2301.05053v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1b4121dc7c9149bb9905f90c6fd46a58d80acb4 --- /dev/null +++ b/zdE4T4oBgHgl3EQfZAyd/content/tmp_files/2301.05053v1.pdf.txt @@ -0,0 +1,86 @@ +arXiv:2301.05053v1 [math.RT] 12 Jan 2023 +G-CIRCULANT MATRICES AND THE CLASSICAL MASCHKE +THEOREM +JON MERZEL +Abstract. In this note, we use the isomorphism of the ring of G-circulant matrices over +a field k with the group ring k[G] to derive a very short proof of the Classical Maschke +Theorem. +Dedicated to J´an Min´aˇc for his 70th birthday +By “the Classical Maschke Theorem” we mean: +Theorem. Let k be a field, G a finite group whose order n is not divisible by the charac- +teristic of k. Then the group ring k[G] is semisimple. +Of course, as k[G] is finite dimensional over k, one needs to show only that the Jacobson +radical rad(k[G]) = 0. +We have the following basic fact about the Jacobson radical (see +e.g. [2], Theorem 4.12): If R is a left artinian ring, then rad(R) is a nil (in fact nilpotent) +ideal. (An ideal I is nil if each element of I is nilpotent; I is nilpotent if Ik = 0 for some +positive integer k.) Of course, this is then the case when R is the group algebra k[G], k a +field, G a finite group. +What we actually need here is that each element of rad(k[G]) is nilpotent; to keep things +elementary, we give an ad hoc proof of a (sufficiently general) special case of the cited result. +Let R be a finite dimensional (associative) algebra over the field k, and α ∈ rad(R). Then +u+αb is a unit in R for any b ∈ R and any unit u ∈ R, so if f(x) ∈ k[x] and f(0) ̸= 0 then +f(α) is a unit in R. Since R is finite dimensional, g(α) = 0 for some nonzero polynomial +g(x) ∈ k[x]; write g(x) = xmf(x) with f(0) ̸= 0. +As f(α) is a unit, we must have m > 0 +and αm = 0, so α is nilpotent. +We now give a brief explicit description of the right regular representation of G. +Date: January 13, 2023. +2020 Mathematics Subject Classification. 15A18, 93C73. +Key words and phrases. Rank-one perturbation, eigenspectra, matrix theory. +1 + +Definition. Let G = {gi}n +i=1 be a finite group written multiplicatively, and k a field. For +each g ∈ G we define the n × n matrix Ag ∈ Mn,n(k) by Ag +i,j = δgig,gj (Kronecker δ). +Simple computations show that Ag is a permutation matrix, that the matrices Ag1, · · · , Agn +are linearly independent (their nonzero entries occur in pairwise disjoint sets of indices), +and that AgAh = Agh. +Thus {Ag| g ∈ G} under matrix multiplication forms a group +isomorphic to G and {� +i +aiAgi| ai ∈ k} is a k-algebra isomorphic to k[G]. +Remark. The matrices in this k-algebra are exactly what are called G-circulant matrices +with entries in k (with respect to the ordering g1, · · · , gn), alternately characterized by +the condition Ai,j = Ak,l whenever g−1 +i gj = g−1 +k gl. +(When G = ⟨g⟩ is the cyclic group of +order n with ordering 1, g, g2, · · · , gn−1, a G-circulant matrix is just what is usually called +a circulant matrix.) See for example Section 2 of [1] (in which the matrix Ag is denoted +P ′ +g). +We now prove the Maschke theorem: +Proof. For convenience, order G so that g1 = 1 (the identity element of G). +Suppose +0 ̸= +n� +i=1 +aigi ∈ rad(k[G]). +We can assume without loss of generality that a1 ̸= 0. (If, say, +aj ̸= 0 we can multiply by g−1 +j .) +Let B = +n� +i=1 +aiAgi be the corresponding G-circulant +matrix, and note that all entries of the principal diagonal of B are a1. +Of course, since +n� +i=1 +aigi is nilpotent, so is B. +But the coefficient of xn−1 in the characteristic polynomial +det(xI − B) of B is −trace(B) = −na1 ̸= 0 since a1 ̸= 0 ̸= n ∈ k. It follows that the +matrix B cannot be nilpotent, a contradiction. Thus there are no nonzero elements of +rad(k[G]). +□ +References +[1] Chebolu, S., Merzel, J., Min´aˇc, J., Muller, L., Nguyen, T. T., Pasini, F. and Tˆan, N. D., 2022. On +the Joins of Group Rings. arXiv preprint arXiv:2208.07413. +[2] T.Y. Lam, A First Course in Noncommutative Rings, 2nd Edition, Springer-Verlag, 2001. +Department of Mathematics, Soka University of America, 1 University Drive, Aliso +Viejo, CA 92656 +Email address: jmerzel@soka.edu +2 + diff --git a/zdE4T4oBgHgl3EQfZAyd/content/tmp_files/load_file.txt b/zdE4T4oBgHgl3EQfZAyd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..951238b815eff65aa24112d40a262fa63a5ad63d --- /dev/null +++ b/zdE4T4oBgHgl3EQfZAyd/content/tmp_files/load_file.txt @@ -0,0 +1,66 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf,len=65 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='05053v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='RT] 12 Jan 2023 G-CIRCULANT MATRICES AND THE CLASSICAL MASCHKE THEOREM JON MERZEL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' In this note, we use the isomorphism of the ring of G-circulant matrices over a field k with the group ring k[G] to derive a very short proof of the Classical Maschke Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Dedicated to J´an Min´aˇc for his 70th birthday By “the Classical Maschke Theorem” we mean: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Let k be a field, G a finite group whose order n is not divisible by the charac- teristic of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Then the group ring k[G] is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Of course, as k[G] is finite dimensional over k, one needs to show only that the Jacobson radical rad(k[G]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' We have the following basic fact about the Jacobson radical (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' [2], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='12): If R is a left artinian ring, then rad(R) is a nil (in fact nilpotent) ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' (An ideal I is nil if each element of I is nilpotent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' I is nilpotent if Ik = 0 for some positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=') Of course, this is then the case when R is the group algebra k[G], k a field, G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' What we actually need here is that each element of rad(k[G]) is nilpotent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' to keep things elementary, we give an ad hoc proof of a (sufficiently general) special case of the cited result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Let R be a finite dimensional (associative) algebra over the field k, and α ∈ rad(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Then u+αb is a unit in R for any b ∈ R and any unit u ∈ R, so if f(x) ∈ k[x] and f(0) ̸= 0 then f(α) is a unit in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Since R is finite dimensional, g(α) = 0 for some nonzero polynomial g(x) ∈ k[x];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' write g(x) = xmf(x) with f(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' As f(α) is a unit, we must have m > 0 and αm = 0, so α is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' We now give a brief explicit description of the right regular representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' 15A18, 93C73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Rank-one perturbation, eigenspectra, matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' 1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Let G = {gi}n i=1 be a finite group written multiplicatively, and k a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' For each g ∈ G we define the n × n matrix Ag ∈ Mn,n(k) by Ag i,j = δgig,gj (Kronecker δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Simple computations show that Ag is a permutation matrix, that the matrices Ag1, · · · , Agn are linearly independent (their nonzero entries occur in pairwise disjoint sets of indices), and that AgAh = Agh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Thus {Ag| g ∈ G} under matrix multiplication forms a group isomorphic to G and {� i aiAgi| ai ∈ k} is a k-algebra isomorphic to k[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' The matrices in this k-algebra are exactly what are called G-circulant matrices with entries in k (with respect to the ordering g1, · · · , gn), alternately characterized by the condition Ai,j = Ak,l whenever g−1 i gj = g−1 k gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' (When G = ⟨g⟩ is the cyclic group of order n with ordering 1, g, g2, · · · , gn−1, a G-circulant matrix is just what is usually called a circulant matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=') See for example Section 2 of [1] (in which the matrix Ag is denoted P ′ g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' We now prove the Maschke theorem: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' For convenience, order G so that g1 = 1 (the identity element of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Suppose 0 ̸= n� i=1 aigi ∈ rad(k[G]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' We can assume without loss of generality that a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' (If, say, aj ̸= 0 we can multiply by g−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=') Let B = n� i=1 aiAgi be the corresponding G-circulant matrix, and note that all entries of the principal diagonal of B are a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Of course, since n� i=1 aigi is nilpotent, so is B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' But the coefficient of xn−1 in the characteristic polynomial det(xI − B) of B is −trace(B) = −na1 ̸= 0 since a1 ̸= 0 ̸= n ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' It follows that the matrix B cannot be nilpotent, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Thus there are no nonzero elements of rad(k[G]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' □ References [1] Chebolu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=', Merzel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=', Min´aˇc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=', Muller, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=', Nguyen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=', Pasini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' and Tˆan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' On the Joins of Group Rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='07413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Lam, A First Course in Noncommutative Rings, 2nd Edition, Springer-Verlag, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content=' Department of Mathematics, Soka University of America, 1 University Drive, Aliso Viejo, CA 92656 Email address: jmerzel@soka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} +page_content='edu 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfZAyd/content/2301.05053v1.pdf'} diff --git a/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf b/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4dc266cc8fffeadb52bee01c446d25bbfd85ef36 --- /dev/null +++ b/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b2fbae99db0c430d846cdb3644ecad32a6ee0eeae217e580a2785682f07131de +size 21128520